In [8] Liang and Bai have shown that the - 4 kC snake graph is an odd harmonious graph for each k ³ 1.
In this paper we generalize this result on cycles by showing that the - n kC snake with string 1,1,…,1 when
n º 0 (mod 4) are odd harmonious graph. Also we show that the - 4 kC snake with m-pendant edges for
each k,m ³ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 k mD -
snake are odd harmonious for each k,m ³ 1 . Finally we present some examples to illustrate the proposed
theories.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ³ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for allm³ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n³ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
We consider here k-valent plane and toroidal maps with faces of size a and b. The faces are said to be in a lego if the faces are organized in blocks that then tile the sphere. We expose some enumeration results and the technical enumeration methods.
Then we expose how we managed to draw the graphs from the combinatorial data.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works
attacked (affected) this problem. Among these works, we find the modelling of the network ad hoc in the
form of a graph. We can resume the problem of coherence of the network ad hoc of a problem of allocation
of frequency
We study a new class of graphs, the fat-extended P4 graphs, and we give a polynomial time algorithm to
calculate the Grundy number of the graphs in this class. This result implies that the Grundy number can be
found in polynomial time for many graphs
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
Applications and Properties of Unique Coloring of GraphsIJERA Editor
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
In [8] Liang and Bai have shown that the kC4 − snake graph is an odd harmonious graph for each k ≥ 1. In this paper we generalize this result on cycles by showing that the kCn − snake with string 1,1,…,1 when n ≡ 0 (mod 4) are odd harmonious graph. Also we show that the kC4 − snake with m-pendant edges for each k,m ≥ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 m∆k - snake are odd harmonious for each k,m ≥ 1 . Finally we present some examples to illustrate the proposed theories.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
FURTHER RESULTS ON ODD HARMONIOUS GRAPHSFransiskeran
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ≥ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for all m ≥ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n≥ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
We consider here k-valent plane and toroidal maps with faces of size a and b. The faces are said to be in a lego if the faces are organized in blocks that then tile the sphere. We expose some enumeration results and the technical enumeration methods.
Then we expose how we managed to draw the graphs from the combinatorial data.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
THE RESULT FOR THE GRUNDY NUMBER ON P4- CLASSESgraphhoc
Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works
attacked (affected) this problem. Among these works, we find the modelling of the network ad hoc in the
form of a graph. We can resume the problem of coherence of the network ad hoc of a problem of allocation
of frequency
We study a new class of graphs, the fat-extended P4 graphs, and we give a polynomial time algorithm to
calculate the Grundy number of the graphs in this class. This result implies that the Grundy number can be
found in polynomial time for many graphs
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ³ 2, n ³ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
Applications and Properties of Unique Coloring of GraphsIJERA Editor
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
In [8] Liang and Bai have shown that the kC4 − snake graph is an odd harmonious graph for each k ≥ 1. In this paper we generalize this result on cycles by showing that the kCn − snake with string 1,1,…,1 when n ≡ 0 (mod 4) are odd harmonious graph. Also we show that the kC4 − snake with m-pendant edges for each k,m ≥ 1 , (for linear case and for general case). Moreover, we show that, all subdivision of 2 m∆k - snake are odd harmonious for each k,m ≥ 1 . Finally we present some examples to illustrate the proposed theories.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
FURTHER RESULTS ON ODD HARMONIOUS GRAPHSFransiskeran
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ≥ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for all m ≥ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n≥ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
New Classes of Odd Graceful Graphs - M. E. Abdel-AalGiselleginaGloria
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles graphhoc
In 1991, Gnanajothi [4] proved that the path graph n
P with n vertex and n −1edge is odd graceful, and
the cycle graph Cm with m vertex and m edges is odd graceful if and only if m even, she proved the
cycle graph is not graceful if m odd. In this paper, firstly, we studied the graphCm∪Pn when m = 4, 6,8,10
and then we proved that the graphCm∪Pn
is odd graceful if m is even. Finally, we described an
algorithm to label the vertices and the edges of the vertex set ( ) m n
V C ∪P and the edge set ( ) m n
E C ∪P .
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
ODD GRACEFULL LABELING FOR THE SUBDIVISON OF DOUBLE TRIANGLES GRAPHSijscmcj
The aim of this paper is to present some odd graceful graphs. In particular we show that an odd graceful labeling of the all subdivision of double triangular snakes ( 2∆k -snake ). We also prove that the all subdivision of 2 m∆1-snake are odd graceful. Finally, we generalize the above two results (the all subdivision of 2 m∆k -snake are odd graceful).
EVEN GRACEFUL LABELLING OF A CLASS OF TREESFransiskeran
A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that
induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a
graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy,
the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.
Group {1, −1, i, −i} Cordial Labeling of Product Related GraphsIJASRD Journal
Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.
K-means Clustering Algorithm with Matlab Source codegokulprasath06
K-means algorithm
The most common method to classify unlabeled data.
Also Checkout: http://bit.ly/2Mub6xP
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VOLUME-7 ISSUE-8, AUGUST 2019 , International Journal of Research in Advent Technology (IJRAT) , ISSN: 2321-9637 (Online) Published By: MG Aricent Pvt Ltd
This article is interested to a detailed computation of the commutators of the Hopaf
algebra Uq(sl(n)). It can be treated as a second way to computation the brackets
of the Hopf algebra Uq(sl(n)) which could be introducing and understanding the
Uq(sl(n)) for the researchers.
Similar to ODD HARMONIOUS LABELINGS OF CYCLIC SNAKES (20)
ON THE PROBABILITY OF K-CONNECTIVITY IN WIRELESS AD HOC NETWORKS UNDER DIFFER...graphhoc
We compare the probability of k-Connectivity of an ad hoc network under Random Way Point (RWP),City Section and Manhattan mobility models. A Network is said to be k Connected if there exists at least k edge disjoint paths between any pair of nodes in that network at any given time and velocity. Initially, for each of the three mobility models, the movement of the each node in the ad hoc network at a given velocity and time are captured and stored in the Node Movement Database (NMDB). Using the movements in the NMDB, the location of the node at a given time is computed and stored in the Node
Location Database (NLDB).
The Impact of Data Replication on Job Scheduling Performance in Hierarchical ...graphhoc
In data-intensive applications data transfer is a primary cause of job execution delay. Data access time depends on bandwidth. The major bottleneck to supporting fast data access in Grids is the high latencies of Wide Area Networks and Internet. Effective scheduling can reduce the amount of data transferred across the internet by dispatching a job to where the needed data are present. Another solution is to use a data replication mechanism. Objective of dynamic replica strategies is reducing file access time which leads to reducing job runtime. In this paper we develop a job scheduling policy and a dynamic data replication strategy, called HRS (Hierarchical Replication Strategy), to improve the data access efficiencies. We study our approach and evaluate it through simulation. The results show that our algorithm has improved 12% over the current strategies
Impact of Mobility for Qos Based Secure Manet graphhoc
Secure multicast communication in Mobile Adhoc Networks (MANETs) is challenging due to its inherent characteristics of infrastructure-less architecture with lack of central authority, limited resources such as bandwidth, energy and power. Several group oriented applications over MANETs create new challenges to routing protocols in terms of QOS requirements. In many multicast interactions, due to its frequent node mobility, new member can join and current members can leave at a time. It is necessary to choose a routing protocol which establishes true connectivity between the mobile nodes. The pattern of movement of members is classified into different mobility models and each one has its own distinct features. It is a crucial part in the performance of MANET. Hence key management is the fundamental challenge in achieving secure communication using multicast key distribution for mobile adhoc networks. This paper describes the impact of mobility models for the performance of a new cluster-based multicast tree algorithm with destination sequenced distance vector routing protocol in terms of QOS requirements such as end to end delay, energy consumption and key delivery ratio. For simulation purposes, three mobility models are considered. Simulation results illustrate the performance of routing protocol with different mobility models and different mobility speed under varying network conditions.
A Transmission Range Based Clustering Algorithm for Topology Control Manetgraphhoc
This paper presents a novel algorithm for clustering of nodes by transmission range based clustering (TRBC).This algorithm does topology management by the usage of coverage area of each node and power management based on mean transmission power within the context of wireless ad-hoc networks. By reducing the transmission range of the nodes, energy consumed by each node is decreased and topology is formed. A new algorithm is formulated that helps in reducing the system power consumption and prolonging the battery life of mobile nodes. Formation of cluster and selection of optimal cluster head and thus forming the optimal cluster taking weighted metrics like battery life, distance, position and mobility is done based on the factors such as node density, coverage area, contention index, required and current node degree of the nodes in the clusters
A Battery Power Scheduling Policy with Hardware Support In Mobile Devices graphhoc
A major issue in the ad hoc networks with energy constraints is to find ways that increase their lifetime. The use of multihop radio relaying requires a sufficient number of relaying nodes to maintainnetwork connectivity. Hence, battery power is a precious resource that must be used efficiently in order to avoid early termination of any node. In this paper, a new battery power scheduling policy based on dynamic programming is proposed for mobile devices.This policy makes use of the state information of each cell provided by the smart battery package and uses the strategy of dynamic programming to optimally satisfy a request for power. Using extensive simulation it is proved that dynamic programming based schedulingpolicyimproves the lifetime of the mobile nodes.Also a hardware support is proposed to succeeds in distinguishing between real-time and non-real-time traffic and provides the appropriate grade of service, to meet the time constraints associated with real time traffic.
A Review of the Energy Efficient and Secure Multicast Routing Protocols for ...graphhoc
This paper presents a thorough survey of recent work addressing energy efficient multicast routing protocols and secure multicast routing protocols in Mobile Ad hoc Networks (MANETs). There are so many issues and solutions which witness the need of energy management and security in ad hoc wireless networks. The objective of a multicast routing protocol for MANETs is to support the propagation of data from a sender to all the receivers of a multicast group while trying to use the available bandwidth efficiently in the presence of frequent topology changes. Multicasting can improve the efficiency of the wireless link when sending multiple copies of messages by exploiting the inherent broadcast property of wireless transmission. Secure multicast routing plays a significant role in MANETs. However, offering energy efficient and secure multicast routing is a difficult and challenging task. In recent years, various multicast routing protocols have been proposed for MANETs. These protocols have distinguishing features and use different mechanisms.
Case Study On Social Engineering Techniques for Persuasion Full Text graphhoc
There are plenty of security software in market; each claiming the best, still we daily face problem of viruses and other malicious activities. If we know the basic working principal of such malware then we can very easily prevent most of them even without security software. Hackers and crackers are experts in psychology to manipulate people into giving them access or the information necessary to get access. This paper discusses the inner working of such attacks. Case study of Spyware is provided. In this case study, we got 100% success using social engineering techniques for deception on Linux operating system, which is considered as the most secure operating system. Few basic principal of defend, for the individual as well as for the organization, are discussed here, which will prevent most of such attack if followed.
Breaking the Legend: Maxmin Fairness notion is no longer effective graphhoc
In this paper we analytically propose an alternative approach to achieve better fairness in scheduling mechanisms which could provide better quality of service particularly for real time application. Our proposal oppose the allocation of the bandwidth which adopted by all previous scheduling mechanism. It rather adopt the opposition approach be proposing the notion of Maxmin-charge which fairly distribute the congestion. Furthermore, analytical proposition of novel mechanism named as Just Queueing is been demonstrated
I-Min: An Intelligent Fermat Point Based Energy Efficient Geographic Packet F...graphhoc
Energy consumption and delay incurred in packet delivery are the two important metrics for measuring the performance of geographic routing protocols for Wireless Adhoc and Sensor Networks (WASN). A protocol capable of ensuring both lesser energy consumption and experiencing lesser delay in packet delivery is thus suitable for networks which are delay sensitive and energy hungry at the same time. Thus a smart packet forwarding technique addressing both the issues is thus the one looked for by any geographic routing protocol. In the present paper we have proposed a Fermat point based forwarding technique which reduces the delay experienced during packet delivery as well as the energy consumed for transmission and reception of data packets.
Fault tolerant wireless sensor mac protocol for efficient collision avoidancegraphhoc
In sensor networks communication by broadcast methods involves many hazards, especially collision. Several MAC layer protocols have been proposed to resolve the problem of collision namely ARBP, where the best achieved success rate is 90%. We hereby propose a MAC protocol which achieves a greater success rate (Success rate is defined as the percentage of delivered packets at the source reaching the destination successfully) by reducing the number of collisions, but by trading off the average propagation delay of transmission. Our proposed protocols are also shown to be more energy efficient in terms of energy dissipation per message delivery, compared to the currently existing protocol.
Enhancing qo s and qoe in ims enabled next generation networksgraphhoc
Managing network complexity, accommodating greater numbers of subscribers, improving coverage to support data services (e.g. email, video, and music downloads), keeping up to speed with fast-changing technology, and driving maximum value from existing networks – all while reducing CapEX and OpEX and ensuring Quality of Service (QoS) for the network and Quality of Experience (QoE) for the user. These are just some of the pressing business issues faced by mobileservice providers, summarized by the demand to “achieve more, for less.” The ultimate goal of optimization techniques at the network and application layer is to ensure End-user perceived QoS. The next generation networks (NGN), a composite environment of proven telecommunications and Internet-oriented mechanisms have become generally recognized as the telecommunications environment of the future. However, the nature of the NGN environment presents several complex issues regarding quality assurance that have not existed in the legacy environments (e.g., multi-network, multi-vendor, and multi-operator IP-based telecommunications environment, distributed intelligence, third-party provisioning, fixed-wireless and mobile access, etc.). In this Research Paper, a service aware policy-based approach to NGN quality assurance is presented, taking into account both perceptual quality of experience and technologydependant quality of service issues. The respective procedures, entities, mechanisms, and profiles are discussed. The purpose of the presented approach is in research, development, and discussion of pursuing the end-to-end controllability of the quality of the multimedia NGN-based communications in an environment that is best effort in its nature and promotes end user’s access agnosticism, service agility, and global mobility
Simulated annealing for location area planning in cellular networksgraphhoc
LA planning in cellular network is useful for minimizing location management cost in GSM network. In fact, size of LA can be optimized to create a balance between the LA update rate and expected paging rate within LA. To get optimal result for LA planning in cellular network simulated annealing algorithm is used. Simulated annealing give optimal results in acceptable run-time
Secure key exchange and encryption mechanism for group communication in wirel...graphhoc
Secured communication in ad hoc wireless networks is primarily important, because the communication signals are openly available as they propagate through air and are more susceptible to attacks ranging from passive eavesdropping to active interfering. The lack of any central coordination and shared wireless medium makes them more vulnerable to attacks than wired networks. Nodes act both as hosts and routers and are interconnected by Multi- hop communication path for forwarding and receiving packets to/from other nodes. The objective of this paper is to propose a key exchange and encryption mechanism that aims to use the MAC address as an additional parameter as the message specific key[to encrypt]and forward data among the nodes. The nodes are organized in spanning tree fashion, as they avoid forming cycles and exchange of key occurs only with authenticated neighbors in ad hoc networks, where nodes join or leave the network dynamically.
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any centralized administrator also the wireless nodes that can dynamically form a network to exchange
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actions communicate with each other path, An ideal choice way the agreement should not only be able to
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paper, the more representative of routing protocols, analysis of individual characteristics and advantages
and disadvantages to collate and compare, and present the all applications or the Possible Service of Ad
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ACTOR GARBAGE COLLECTION IN DISTRIBUTED SYSTEMS USING GRAPH TRANSFORMATIONgraphhoc
A lot of research work has been done in the area of Garbage collection for both uniprocessor and
distributed systems. Actors are associated with activity (thread) and hence usual garbage collection
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transform the active reference graph into a graph which captures all the features of actors and looks like
passive reference graph then any passive reference graph algorithm can be applied for it. But the cost of
transformation and optimization are the core issues. An attempt has been made to walk through these
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A Proposal Analytical Model and Simulation of the Attacks in Routing Protocol...graphhoc
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The Neighborhood Broadcast Problem in Wireless Ad Hoc Sensor Networksgraphhoc
to all neighbors of a network node v under the assumption that v does not participate due to
being corrupted or damaged. We present practical network protocol that can be used completely
reactive. It is parameterized with a positive integer k ∈ N and it is proven to guarantee delivery for
k ≥ 2d−1, if node v is d-locally connected, which means that the set of nodes with distance between
1 and d to v induces a connected subgraph of the communication graph. It is also shown that the
number of participating nodes is optimal under the restriction to 1-hop neighborhood information.
The protocol is also analyzed in simulations that demonstrate very high success rates for very low
values of k.
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ODD HARMONIOUS LABELINGS OF CYCLIC SNAKES
1. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
DOI : 10.5121/jgraphoc.2013.5301 1
ODD HARMONIOUS LABELINGS OF CYCLIC
SNAKES
M. E. Abdel-Aal
Department of Mathematics, Faculty of Science,
Benha University, Benha 13518, Egypt
ABSTRACT
In [8] Liang and Bai have shown that the −4kC snake graph is an odd harmonious graph for each 1≥k .
In this paper we generalize this result on cycles by showing that the −nkC snake with string 1,1,…,1 when
)4(mod0≡n are odd harmonious graph. Also we show that the −4kC snake with m-pendant edges for
each 1, ≥mk , (for linear case and for general case). Moreover, we show that, all subdivision of 2 km∆ -
snake are odd harmonious for each 1, ≥mk . Finally we present some examples to illustrate the proposed
theories.
KEYWORDS
Odd harmonious labeling, pendant edges, Cyclic snakes, Subdivision double triangular snakes.
1. INTRODUCTION
Graph labeling is an active area of research in graph theory which has mainly evolved through its
many applications in coding theory, communication networks, mobile telecommunication
system. Optimal circuits layouts or graph decompositions problems, no name just a few of them.
Most graph labeling methods trace their origion to one introduced by Rosa [1] called such a
labeling a β-valuation and Golomb [2] subsequently called graceful labeling, and one introduced
by Graham and Sloane [3] called harmonious labeling. Several infinite families of graceful and
harmonious graphs have been reported. Many illustrious works on graceful graphs brought a tide
to different ways of labeling the elements of graph such as odd graceful.
A graph G of size q is odd-graceful, if there is an injection f from V (G) to {0,1,2,…, 2q -1} such
that, when each edge xy is assigned the label or weight f(x) - f(y) , the resulting edge labels are
{1, 3, 5, . . . , 2q - 1}. This definition was introduced by Gnanajothi [4]. Many researchers have
studied odd graceful labeling. Seoud and Abdel-Aal [5] they determine all connected odd
graceful graphs of order ≤ 6. For a dynamic survey of various graph labeling problems we refer to
Gallian [6].
2. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
2
Throughout this work graph G = (V(G), E(G)) we mean a simple, finite, connected and
undirected graph with p vertices and q edges. For all other standard terminology and notions we
follow Harary [7].
A graph G is said to be odd harmonious if there exists an injection f: V(G) → {0, 1, 2, …, 2q-1}
such that the induced function f*
: E(G) {1, 3, . . . , 2q − 1} defined by f*
(uv) = f(u) + f(v) is a
bijection. Then f is said to be an odd harmonious labeling of G [8].
A graph −nkC snake was introduced by Barrientos [9], as generalization of the concept of
triangular snake introduced by Rosa [10].
Let G be a −nkC snake with 2≥k . Let 121 ...,,, −kuuu be the consecutive cut-vertices of G.
Let id be the distance between iu and 1+iu in G, 21 −≤≤ ki and the string )...,,,( 221 −kddd of
integers. Hence, any graph G = −nkC snake, can be represented by a string. For instance, the
string (from left to right) of the −48C snake on Figure (4) is 2,2,1,2,1,1. Gracefulness of the
kind of −4kC snake studied by Gnanajothi have string 1, 1,…, 1. And the labelings given by
Ruiz considered by −4kC snake with string 2,2 ,…,2. We obtain in Theorem 2.4 an odd
harmonious labelings of the −4kC snake with string 1,1,…,1. If the string of given −nkC snake
is
2
n
,
2
n
,…,
2
n
, we say that −nkC snake is linear.
This paper can be divided into two sections. Section 1, we show that the graphs −4kC snake ʘ
1mk (the graph obtained by joining m-pendant edges to each vertex of −4kC snake 1, ≥mk ) for
linear and general cases of −4kC snake for each 1≥k are odd harmonious. We also obtain an
odd harmonious labeling of −nkC snake with the sequence string is 1, 1,..., 1 and when
)4(mod0≡n . In section 2, we show that, an odd harmonious labeling of the all subdivision of
double triangular snakes ( 2 k∆ -snake). Finally, we prove that the all subdivision of 2 km∆ -
snake are odd harmonious for each 1, ≥mk .
2. MAIN RESULTS
In [8, Corollary 3.2 ] Liang and Bai in Corollary 3.2 (2) when i =1, they have shown that, the
−4kC snake graph is an odd harmonious graph for each 1≥k . We extended this result to obtain
an odd harmonious labeling for the corona −4kC snake graph (the graph obtained by joining m
pendant edges to each vertex −4kC snake,) are denoted by −4kC snake ʘ 1mk .
Theorem 2.1.
The linear graphs −4kC snake ʘ 1mk are odd harmonious for 1, ≥mk
3. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
3
Proof. Consider the linear graph −4kC snake, 1≥k which has the vertices iw , ju , and jv
where ki ,...,2,1,0= , kj ,...,2,1= . In order to get the linear −4kC snake ʘ 1mk , 1, ≥mk , we
add m-pendant edges l
iw ,
l
ju , and
l
jv to each vertex of iw , ju , and jv respectively such that
ml ,...,2,1,0= . Now, let G be the linear −4kC snake ʘ 1mk , 1, ≥mk be described as
indicated in Figure 1.
2v kv1v
1u
0w
1w 2w 1−kw
kw
ku
1
0w 2
0w m
w0
1
1v 2
1v m
v1
1
1w 2
1w m
w1
2u
1
2v 2
2v m
v2
1
2w 2
2w m
w2
1
1u 2
1u m
u1
1
2u 2
2u m
u2
1
ku 2
ku m
ku
1
kw 2
kw m
kw
1
1−kw 2
1−kw
m
kw 1−
1
kv 2
kv m
kv
Figure 1
It is clear that the number of edges of the graph H is q = 4k + m(3k+1) . We define the labeling
function f : V(G) → {0, 1, 2, …, 8k +2 m(3k+1) - 1} as follows:
.0,4)( kiiwf i ≤≤=
,1,34)( kiivf i ≤≤−=
,1,14)( kiiuf i ≤≤−=
mjkiimjqwf j
i ≤≤≤≤++−−= 1,0,1])2([2)(
mjkimimjmkuf j
i ≤≤≤≤+++−−+= 1,1,22)1(42)2(4)(
mjkiimjmkvf j
i ≤≤≤≤−+−−+= 1,1),1)(1(42)2(4)( .
The edge labels will be as follows:
• The vertices 1−iw , iv , ki ≤≤1 , induce the edge labels
78)()( 1 −=+− ivfwf ii , ki ≤≤1 .
• The vertices 1−iw , iu , ki ≤≤1 , induce the edge labels
58)()( 1 −=+− iufwf ii , ki ≤≤1 .
• The vertices iv , iw ki ≤≤1 , induce the edge labels
38)()( −=+ iwfvf ii , ki ≤≤1 .
• The vertices iu , iw ki ≤≤1 , induce the edge labels
18)()( −=+ iwfuf ii , ki ≤≤1 .
The remaining odd edge labels from 8k+1 to 2k+2m(3k+1) are obtained from the following
• .1,1,1242)2(4)()( mjkimmijmkufuf j
jj ≤≤≤≤++−−+=+
• ,1,1,1442)2(4)()( mjkimmijmkvfvf j
jj ≤≤≤≤++−−+=+
4. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
4
• ,1,0,1][2)()( mjkimijqwfwf j
ii ≤≤≤≤+−−=+
So { )(:)()( GEuvvfuf ∈+ }= {1,3,5,…, 2q-1}. Hence the graph G is odd harmonious.
Example 2.3. An odd harmonious labeling of the graph linear −43C snake ʘ 12k , is shown in
Figure (2).
Figure (2): The graph linear −43C snake ʘ 12k with its odd harmonious labeling.
For the general form of cyclic graph we obtain the following result.
Theorem 2.2.
The following graphs are odd harmonious
(i) −4kC snake for each 1≥k ,
(ii) −4kC snake ʘ 1mk for each 1, ≥mk , (the general form).
Proof.
The graph −4kC snake can be considered as a bipartite graph (one partite set has black vertices
and the other has white vertices) it is possible to embed it, on a square grid as is showed in the
next Figure 3.
Figure 3
Let G be the graph −4kC snake ʘ 1mk which obtained by joining m- pendant edges to each
vertex of −4kC snake. Then )13)(1()( ++= kmGV and )13(4)( ++= kmkGE . Now, we
are running the following steps sequentially in order to prove the Theorem:
Step 1. Since −4kC snake is a bipartite graph so it has one partite set has black vertices and the
other has white vertices as is showed in Figure (3). Put black vertices in a string, ordered by
diagonals from left to right and inside each diagonal from bottom to top, assign to them from an
arithmetic progression of difference 4 which first term is zero, when we move to another diagonal
5. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
5
we use an arithmetic progression of difference 2, counting until the last black vertex has been
numbered. Similarly, Put the white vertices on a string, ordered for diagonals from left to right
and inside each diagonal from top to bottom, starting with the first diagonal assign numbers from
an arithmetic progression of difference 2, which first term is 1, when we move to another
diagonal we use an arithmetic progression of difference 2, counting until the last white vertex has
been numbered.
Step 2. In this step, we are labeling the vertices of m-pendant edges which contact with the
white diagonals, from right to left and inside each white diagonal from bottom to top, assign to
them from an arithmetic progression of difference 2, which first term is z such that z = y + 2
where y is the last vertex labeling of black diagonal, when we move to a new vertex of the white
diagonal, the first vertex of m- pendant edges is labeled by an arithmetic progression of difference
4, but the arithmetic progression of difference 2 has been used with the remain (m-1)vertices of
m-pendant edges. We move from a vertex to another of the white diagonals until the last white
vertex.
Step 3. Finally, we are labeling the vertices of m-pendant edges which contact with the black
diagonals, from left to right and inside each black diagonal from bottom to top, assign to them
from an arithmetic progression of difference (-2), which first term is (2q-1) where q is the size of
G, when to move to a new vertex of the black diagonal, the first vertex of m- pendant edges is
labeled by an arithmetic progression of difference (-6), but the arithmetic progression of
difference(-2) has been used with the remain (m-1)vertices of m-pendant edges. We move from a
vertex to another of the black diagonals until the last black vertex.
Now, we have complete the proof by running the above steps, i.e. we mention only the vertices
labels and the reader can fulfill the proof as we did in the previous theorem where step1 give us
an odd harmonious labeling of the graph −4kC snake for each 1≥k , and step1- step3 give us an
odd harmonious labeling of the graph −4kC snake ʘ 1mk for each 1, ≥mk (in general case).
The following example illustrates the last result.
Example 2.3. An odd harmonious labeling of the graph −45C snake ʘ 12k (for general case) is
shown in Figure 4.
Figure 4: The graph −45C snake ʘ 12k (for general case) with its odd harmonious labeling.
6. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
6
The graphs −nkC snake when the sequence string is (1, 1, 1,…, 1) when )4(mod0≡n
are studied in the following Theorem:
Theorem 2.4. The graphs −mkC4 snake for each k, m ≥ 1, with string (1, 1,…, 1) are odd
harmonious.
Proof. Let G = −mkC4 snake = −− mCn 4)1( snake can be described as indicated in Figure 5
1
1v
2
1v
m
v 2
1
3
1v
12
1
−m
v
1
2v 1
3v
1
1−nv 1
nv
2
21v 2
22v
3
21v 3
22v
12
22
−m
v12
21
−m
v
m
v 2
21
m
v 2
22
2
31v
2
32v
3
32v3
31v
12
31
−m
v 12
32
−m
v
m
v 2
31
m
v 2
32
2
1)1( −nv
3
1)1( −nv
12
1)1(
−
−
m
nv
m
nv 2
1)1( −
2
2)1( −nv
3
2)1( −nv
12
2)1(
−
−
m
nv
m
nv 2
2)1( −
2
1nv
3
1nv
12
1
−m
nv
m
nv 2
1
Figure 5
It is clear that )1(4)( −= nmGE . We define the labeling function
f : V(G)) → {0, 1, 2, …, 8m(n – 1)-1}
as follows:
−=+−
−=−
=
.1,...6,4,2,1)1(4
1,...5,3,1),1(4
)( 1
norniim
norniim
vf i
mjjvf j
21,1)( 1 ≤≤−=
For labeling the vertices mjniv j
i 22,2,1 ≤≤≤≤ we consider the following two
cases:
Case(i)
if i is odd, ni ≤≤3 we have the following labeling, for each mj 22 ≤≤
1)()( 1
1 +−= jvfvf i
j
i
Case(ii)
if i is even, ni ≤≤2 we have the following labeling, for each mj 22 ≤≤ :
=−−
−=+−
=
.2,...6,4,2,1)(
12,...5,3,1,1)(
)(
1
1
1
mjjvf
mjjvf
vf
i
i
j
i
Now we label the remaining vertices mjniv j
i 21,12,2 ≤≤−≤≤ as follows:
.12,21),1(2)()( 1
12 −≤≤≤≤−+= nimjjvfvf i
j
i
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It follows that f admits an odd harmonious labeling for −− mCn 4)1( snake. Hence −− mCn 4)1(
snake for each 1,2 ≥≥ mn with the string (1,1,…,1) are odd harmonious graphs.
Example 2.5. Odd harmonious labeling of graph −84C snake with the string (1,1,…,1) is shown
in Figure 6.
Figure 6: The graph −84C snake with its odd harmonious labeling.
3. SUBDIVISION OF DOUBLE TRIANGLES SNAKE
Rosa [10] defined a triangular snake (or ∆ -snake) as a connected graph in which all blocks are
triangles and the block-cut-point graph is a path. Let k∆ -snake be a ∆ -snake with k blocks
while kn∆ -snake be a ∆ -snake with k blocks and every block has n number of triangles with
one common edge.
David Moulton [11] proved that triangular snakes with p triangles are graceful if p is congruent to
0 or 1 modulo 4. Xu [12] proved that they are harmonious if and only if p is congruent to 0, 1 or
3 modulo 4.
A double triangular snake is a graph that formed by two triangular snakes have a common path.
The harmonious labeling of double triangle snake introduced by Xi Yue et al [13]. It is known
that, the graphs which contain odd cycles are not odd harmonious so we used the subdivision
notation for odd cycle in order to prove that all subdivision of double triangular snakes are odd
harmonious.
Theorem 3.1. All subdivision of double triangular snakes ( 2 k∆ -snake 1≥k ) are odd
harmonious.
Proof. Let G = 2 k∆ -snake has q edges and p vertices. The graph G consists of the vertices
( u1, u2,…,uk+1), (v1,v2,…,vk ), ( w1,w2,…,wk ) therefore we get the subdivision of double triangular
snakes S(G) by subdividing every edge of double triangular snakes 2 k∆ -snake exactly once.
Let yi be the newly added vertex between ui and ui+1 while wi1 and wi2 are newly added vertices
between
wi ui and wi ui+1 respectively, where 1 ≤i ≤k. Finally, vi1 and vi2 are newly added vertices between
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vi ui and vi ui+1respectivley, such that 1≤i ≤k. Let the graph S(G) be described as indicated in
Figure 7
kw1w 2w 3w
kv1v 2v 3v
ku 1+ku
1u
2u 3u 4u
11w
12w 21w 22w 31w 32w 1kw
2kw
11v
12v 21v 22v 31v 32v 1kv 2kv
1y 2y 3y ky
Figure 7: the subdivision of double triangular snakes ( 2 k∆ -snake).
It is clear that the number of edges of the graph S(G) is 10k. We define the labeling function:
f : V(S(G)) → {0, 1, 2, 3, …, 20k - 1}
as follows:
f( ui ) = 6( i -1 ) , 1 ≤ i ≤ k + 1= n,
f( yi ) = 14 i -11 , 1 ≤ i ≤ k = n-1,
f( wi ) = 6i +4 , 1 ≤ i ≤ k = n-1,
f( vi ) = 6i -4 , 1 ≤ i ≤ k = n-1,
f( wi j) = 14i + 8j - 21 , 1 ≤ i ≤ k = n-1 , j=1,2,
f( vi j) = 14i + 6j - 15 , 1 ≤ i ≤ k = n-1 , j=1,2.
The edge labels will be as follows:
• The vertices ui and wi1 , 1 ≤ i ≤ k, induce the edge labels {20i-19, 1 ≤ i ≤ k} = {1,21,…,20k-19}.
• The vertices ui and yi , 1 ≤ i ≤ k, induce the edge labels{20i-17, 1 ≤ i ≤ k} = {3,23,…,20k-17}.
• The vertices ui and vi1 , 1 ≤ i ≤ k, induce the edge labels{20i-15, 1 ≤ i ≤ k} = {5,25,…,20k-15}.
• The vertices vi1 and vi , 1 ≤ i ≤ k, induce the edge labels{20i-13, 1 ≤ i ≤ k} = {7,27,…,20k-13}.
• The vertices yi and ui+1 , 1 ≤ i ≤ k, induce the edge labels{20i-11, 1 ≤ i ≤ k} = {9,29,…,20k-11}.
• The vertices wi1 and wi , 1 ≤ i ≤ k, induce the edge labels{20i-9, 1 ≤ i ≤ k} = {11,31 ,…,20k-9}.
• The vertices vi and vi2, 1 ≤ i ≤ k, induce the edge labels{20i-7, 1 ≤ i ≤ k} = {13,33 ,…,20k-7}.
• The vertices wi2 and ui+1 , 1 ≤ i ≤ k, induce the edge labels{20i-5, 1 ≤ i ≤ k} = {15,35 ,…,20k-5}.
• The vertices vi2 and ui+1 , 1 ≤ i ≤ k, induce the edge labels{20i-3, 1 ≤ i ≤ k} = {17,37 ,…,20k-3}.
• The vertices wi and wi2, 1 ≤ i ≤ k, induce the edge labels{20i-1, 1 ≤ i ≤ k} = {19,39 ,…,20k-1}.
So we obtain all the edge labels{1,3,5,…,20k-1}. Hence the subdivision of double triangular
snakes ( 2 k∆ -snake , 1≥k ) are odd harmonious.
Theorem 3.2.
All subdivision of 2 km∆ -snake, m, k 1 are odd harmonious.
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Proof. Let G = 2 km∆ -snake has q edges and p vertices. The graph G consists of the vertices (
u1, u2,…,uk+1), )...,,,( 21 m
iii vvv , kiwww m
iii ≤≤1),...,,,( 21
. Therefore we get generalized the
subdivision of double triangular snakes S(G) by subdividing every edge of 2 km∆ -snake exactly
once. Let y1 be the newly added vertex between iu and 1+iu while 1
j
iw and 2
j
iw are newly added
vertices between ui
j
iw and 1+i
j
i uw respectively. Finally , 1
j
iv and 2
j
iv are newly added vertices
between ui
j
iv and j
iv ui+1 respectively where i = 1, 2, . . ., k and j = 1, 2, 3, . . .,m ( Figure 8 ). It
is Clear that, the number of edges of the graph S(G) is q = k (8m + 2) edges.
1
1v
1y 2y 1−ny
2
1v
m
v1
1
2v
2
2v
m
v 2
1
1−nv
2
1−nv
m
nv 1−
1
1w
2
1w
m
w1
m
w 2
2
2w
1
2w 1
1−nw
2
1−nw
m
nw 1−
1u 2u 3u 1−nu
nu
1
11w
1
12w
2
11w 2
12w
m
w11
m
w12
1
21w
1
22w
2
21w
2
22w
m
w 21
m
w 22
1
1)1( −nw 1
2)1( −nw
2
1)1( −nw 2
2)1( −nw
m
nw 1)1( −
m
nw 2)1( −
1
11v
1
12v
2
11v
m
v11
2
12v
m
v12
1
21v
1
22v
2
21v
m
v 21
2
22v
m
v 22
m
nv 1)1( −
m
nv 2)1( −
1
1)1( −nv
2
1)1( −nv
2
2)1( −nv
1
2)1( −nv
Figure 8: the subdivision of 2 km∆ -snake
We define the labeling function f : V(S(G)) → {0, 1, 2, 3, …, 2k(8m+2) - 1}
as follows:
f( ui ) = (4m + 2)( i -1 ) , 1 ≤ i ≤ k + 1= n,
f( yi ) = (12m + 2) i - 10m - 1 , 1< i < k ,
f( l
iv ) = (4m + 2)(i - 1) + 4l - 2 , 1 < i < k , 1 < l < m ,
f( j
iw ) =(4m + 6) + (4m + 2)(i - 1) +4(j - 1) , 1 < i < k , 1 < j< m ,
f( j
ilw ) = (2m-1)+(12m+2)(i - 1) + (4m + 2)(l- 1) - 2(j - 1), 1 < i < k , 1 < l < 2, 1 < j < m ,
f( j
ilv ) = (4m + 1) + (12m + 2)(i - 1) + (8 m + 2)(l - 1) - 2(j - 1), 1 < i < k , 1 < l < 2, 1 < j
< m,
f( 1
ilv ) =(4m + 1) + (12m + 2)(i - 1) + (6 m + 2)(l - 1), 1 < i < k , 1 < l< 2.
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In a view of the above defined labeling pattern f is odd harmonious for the graph S(G). Hence S(
2 km∆ -snake) is odd-graceful for all m >1, k > 1.
Illustration 3.3. An odd harmonious labeling of the graph subdivision of 36∆ -snake is shown in
Figure 9.
7 45 83
1
3
5
15
17
19
9
11
13
35
37
33
39
41
43
47
49
51
53
55
57
73
75
71
77
79
81
85
87
89
91
93
95
111
113
109
14 28
42
26
22
18
10
6
2 16
20
24
32
36
40 54
50
46
30
34
38
Figure 9: the graph subdivision of 36∆ -snake with its odd harmonious labeling.
4. CONCLUSION
Harmonious and odd harmonious of a graph are two entirely different concepts. A graph may
posses one or both of these or neither. In the present work we investigate several families of odd
harmonious cyclic snakes. To investigate similar results for other graph families and in the
context of different labeling techniques is open area of research.
5. REFERENCES
[1] A. Rosa, (1967) “On certain valuations of the vertices of a graph, in Theory of Graphs”, International
Symposium, Rome, July 1966, Gordon and Breach, NewYork and Dunod, Paris, pp. 349–355.
[2] S.W. Golomb, (1972) “How to number a graph, in Graph Theory and Computing”, R.C. Read,
ed.,Academic Press, NewYork, pp. 23–37.
[3] R. L. Graham, N. J. A. Sloane, (1980) “On additive bases and harmonious graphs”, SIAM J. Algebr.
Disc. Math., Vol 1, No 4, pp. 382-404.
[4] R.B. Gnanajothi, (1991) “Topics in graph theory”, Ph.D. thesis, Madurai Kamaraj University, India.
[5] M.A. Seoud and M.E. Abdel-Aal, (2013) On odd graceful graphs, Ars Combin., 108, pp. 161-185.
[6] J. A. Gallian. (2011) “A Dynamic Survey of Graph Labeling”, Electronic J. Combin. Fourteenth
edition.
[7] F. Harary. (1969) “ GpaphTheory”, Addison-Wesley, Reading MA.
11. International journal on applications of graph theory in wireless ad hoc networks and sensor networks
(GRAPH-HOC) Vol.5, No.3, September 2013
11
[8] Z. Liang, Z . Bai. (2009) On the odd harmonious graphs with applications, J. Appl.Math. Comput.,
29, pp.105-116.
[9] C. Barrientos, (2004) “Difference Vertex Labelings”, Ph.D.Thesis, University Politecnica De
Cataunya, Spain.
[10] A. Rosa, (1967) “Cyclic Steiner Triple System and Labelings of Triangular Cacti”, Scientia, 5, pp.
87-95.
[11] D. Moulton, (1989) “Graceful labelings of triangular snakes”, Ars Combin., 28, pp. 3-13.
[12] S.D. Xu, (1995) “ Harmonicity of triangular snakes”, J. Math. Res. Exposition, 15, pp. 475-476
[13] Xi Yue, Yang Yuansheng and Wang Liping, (2008 ) “On harmonious labeling of the double
triangular n snake”, Indian J. pure apple. Math.,39(2), pp. 177-184.
Author
Mohamed Elsayed Abdel-Aal received the B.Sc. (Mathematics) the M.Sc.(Pure
Mathematics-Abstract Algebra) degree from Benha University, Benha, Egypt in 1999,
2005 respectively. Also, he received Ph.D. (Pure Mathematics) degree from Faculty of
Mathematics, Tajik National University, Tajikistan, in 2011. He is a University lecturer
of Pure Mathematics with the Benha University, Faculty of Science, Department of
Pure Mathematics. His current research is Ordinary –partial differential equations,
Graph Theory and Abstract Algebra.