This document provides an overview of the CS304 "Algorithmic Combinatorics 2" course for third year computer science students. The course covers graph theory and constructive algorithmic combinatorics. It will assess students through oral examination, class work involving writing computer programs, homework, and a final examination. The class will learn about graph concepts like vertices, edges, and degrees as well as graph types, examples, and basic properties. Students are expected to treat each other and the instructor with respect and complete all assessments honestly.
A graph is a diagram displaying data which show the relationship between two or more quantities, measurements or indicative numbers that may or may not have a specific mathematical formula relating them to each other.
This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
A graph is a diagram displaying data which show the relationship between two or more quantities, measurements or indicative numbers that may or may not have a specific mathematical formula relating them to each other.
This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
RETOOLING OF COLOR IMAGING IN THE QUATERNION ALGEBRAmathsjournal
A novel quaternion color representation tool is proposed to the images and videos efficiently. In this work, we consider a full model for representation and processing color images in the quaternion algebra. Color images are presented in the threefold complex plane where each color component is described by a complex image. Our preliminary experimental results show significant performance improvements of the proposed approach over other well-known color image processing techniques. Moreover, we have shown how a particular image enhancement of the framework leads to excellent color enhancement (better than other algorithms tested). In the framework of the proposed model, many other color processing algorithms, including filtration and restoration, can be expressed.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
RETOOLING OF COLOR IMAGING IN THE QUATERNION ALGEBRAmathsjournal
A novel quaternion color representation tool is proposed to the images and videos efficiently. In this work, we consider a full model for representation and processing color images in the quaternion algebra. Color images are presented in the threefold complex plane where each color component is described by a complex image. Our preliminary experimental results show significant performance improvements of the proposed approach over other well-known color image processing techniques. Moreover, we have shown how a particular image enhancement of the framework leads to excellent color enhancement (better than other algorithms tested). In the framework of the proposed model, many other color processing algorithms, including filtration and restoration, can be expressed.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
All Things Open 2014 - Day 2
Thursday, October 23rd, 2014
Phil Shapiro
Public Geek for Takoma Park Maryland Library
Open Gov/Data
Open Sourcing the Public Library
mapa conceptual Gerencia de Proyectos describe el rol del profesional de proyectos y el ciclo de vida de proyectos, responsables y planeación estratégica.
Lecture #2 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
Medical Conferences, Pharma Conferences, Engineering Conferences, Science Conferences, Manufacturing Conferences, Social Science Conferences, Business Conferences, Scientific Conferences Malaysia, Thailand, Singapore, Hong Kong, Dubai, Turkey 2014 2015 2016
Global Research & Development Services (GRDS) is a leading academic event organizer, publishing Open Access Journals and conducting several professionally organized international conferences all over the globe annually. GRDS aims to disseminate knowledge and innovation with the help of its International Conferences and open access publications. GRDS International conferences are world-class events which provide a meaningful platform for researchers, students, academicians, institutions, entrepreneurs, industries and practitioners to create, share and disseminate knowledge and innovation and to develop long-lasting network and collaboration.
GRDS is a blend of Open Access Publications and world-wide International Conferences and Academic events. The prime mission of GRDS is to make continuous efforts in transforming the lives of people around the world through education, application of research and innovative ideas.
Global Research & Development Services (GRDS) is also active in the field of Research Funding, Research Consultancy, Training and Workshops along with International Conferences and Open Access Publications.
International Conferences 2014 – 2015
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Benefits Of Innovative 3d Graph Techniques In Construction IndustryA Makwana
The construction industry is the second largest industry of the country after agriculture. It makes a significant contribution to the national economy and provides employment to a large number of people. In its path of advancement, the industry has to overcome a number of challenges. One of the challenges is effective utilization of available technology. 3D Graph is one such technique that is rarely being used. Graphs and plots are a natural way to visualize data. It hardly needs saying that their use is common even in non-technical documents. Unfortunately, much work is often required (and rarely performed) to produce plots with sufficient output quality to match a well-typeset document. Recent years have witnessed a rapid development in the technologies that are related to digital visualization and simulation together with the technologies that try to link between digital and physical modelling. Many engineering and design practitioners have begun to apply selective technologies in their practices. The research attempts to classify the possible technologies that can be used throughout stages of urban design projects according to their purpose.
Graph Summarization with Quality GuaranteesTwo Sigma
Given a large graph, the authors we aim at producing a concise lossy representation (a summary) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation.
Comparative Analysis of Algorithms for Single Source Shortest Path ProblemCSCJournals
The single source shortest path problem is one of the most studied problem in algorithmic graph theory. Single Source Shortest Path is the problem in which we have to find shortest paths from a source vertex v to all other vertices in the graph. A number of algorithms have been proposed for this problem. Most of the algorithms for this problem have evolved around the Dijkstra’s algorithm. In this paper, we are going to do comparative analysis of some of the algorithms to solve this problem. The algorithms discussed in this paper are- Thorup’s algorithm, augmented shortest path, adjacent node algorithm, a heuristic genetic algorithm, an improved faster version of the Dijkstra’s algorithm and a graph partitioning based algorithm.
An analysis between exact and approximate algorithms for the k-center proble...IJECEIAES
This research focuses on the k-center problem and its applications. Different methods for solving this problem are analyzed. The implementations of an exact algorithm and of an approximate algorithm are presented. The source code and the computation complexity of these algorithms are presented and analyzed. The multitasking mode of the operating system is taken into account considering the execution time of the algorithms. The results show that the approximate algorithm finds solutions that are not worse than two times optimal. In some case these solutions are very close to the optimal solutions, but this is true only for graphs with a smaller number of nodes. As the number of nodes in the graph increases (respectively the number of edges increases), the approximate solutions deviate from the optimal ones, but remain acceptable. These results give reason to conclude that for graphs with a small number of nodes the approximate algorithm finds comparable solutions with those founds by the exact algorithm.
A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmc
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In
this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected
weighted graph with n vertices and m edges. This algorithm uses the cluster techniques to reduce the
number of processors by fraction 1/f (n) and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an
arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on
EREWPRAM model. In general, the proposed algorithm is the simplest one.
A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmcj
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected weighted graph with vertices and edges. This algorithm uses the cluster techniques to reduce the number of processors by fraction and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on EREWPRAM model. In general, the proposed algorithm is the simplest one.
Lecture #3 for course CS301: "Algorithmic Combinatorics I" for 3rd year students with a computer science major, Faculty of Science, Ain Shams University, Academic Year WS2014/2015.
1. Computer Science Division: CS304
Course CS304, Part:
“ALGORITHMIC COMBINATORICS 2 ”
( 3rd
Year Students )
Prepared by:
Dr. Ahmed AshryAhmed AshryAhmed Ashryhttp://www.researchgate.net/profile/Ahmed_Abdel-Fattah
Lecture #1
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 1 (of 25)
2. Computer Science Division: CS304
Course Description:
Algorithmic Combinatorics 2 consists of 2 subjects:
1 an introductory course on “graph theory” to make the students
familiar with graphs and basic theorems that give well-known,
important characterizations of famous graph types.
(Needed in many courses such as DS, DB, compilers design, parallel algorithms, etc.)
2 “constructive algorithmic combinaorics”, in which we study those
“algorithms” for listing fundamental objects in combinatorics.
(Theoretical study should have been given in 301Comp).
Course Material/References:
1 Text: Chapters 10 and 11 of Rosen, K. (2012) [Rosen, 2012]:
Discrete Mathematics and Its Applications (7th edition). McGraw-Hill.
2 Text: [Tucker, 2012, Bender and Williamson, 2013]
3 Lecture slides, practice assignments, and other supplementary material:
Disseminated over the internet.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 2 (of 25)
3. Computer Science Division: CS304
Organizational Remarks1
1Image source: http://plus.google.com/+GawishElwarka/posts
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 3 (of 25)
4. Computer Science Division: CS304
Assessment: “Algorithmic Combinatorics 2” is worth 50% of CS304
Oral Examination (4% of CS304) [≡ 5 Marks]
Class Work (a.k.a. “Mid-term”) (16% of CS304) [≡ 20 Marks]
Final Examination (30% of CS304) [≡ 37.5 Marks]
Total (50% of CS304) [≡ 62.5 Marks]
Class Work is the new “Mid-Term”:
You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!You will end up writing THREE computer programs!
1 Form teams of 2 to 4 students.
2 Select as many programs from the list in [Rosen, 2012, Pages 742 and 808].
Deadlines & Constraints:
Fri., 19-Feb.-2016 (between 10:00pm and 10:59pm CLT) Team formations &
program selections/assignments. Each 10 minutes of earlier
suggestions decrease 0.5 mark of your expected total.
Fri., 01-April-2016 (11:59pm CLT) Delivery of codes. Each 180 minutes of later
submissions decrease 0.5 mark of your expected total.
As usual, you can still earn bonus marks from HWs (6% max.) [≡ 7.5 Marks]
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 4 (of 25)
5. Computer Science Division: CS304
Codes of Honor
I, Ahmed Ashry, pledge my honor that I will:
1 Present the course material as correctly and completely as possible.
2 Answer all questions to the best of my knowledge.
3 Assess students fairly.
4 Treat everyone with respect.
5 Be there for everyone with academic support and experience
sharing.
6 Be always on time or apologize in advance, if I physically can.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 6 (of 25)
6. Computer Science Division: CS304
Codes of Honor
I, 3rd Year Student: Your Name, pledge my honor that I will:
1 Do my best to learn the material presented in class.
2 Help myself by attempting to solve (additional) homework.
3 Help my fellow students with course material as much as I can.
4 Not receive nor give unauthorized assistance to fellow students in
exams.
5 Treat everyone with respect.
6 Be always on time (or apologize in advance, if I physically can).
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 6 (of 25)
7. Computer Science Division: CS304
“Graphs”: A Motivational Question
The Problem of Supplying 3 Houses with 3 Utilities.2
2For this and other motivational questions in graph theory, see next lectures.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 7 (of 25)
8. Computer Science Division: CS304
“Graphs”: Meet the Concept
Conceptually, a graph consists of “points” and their connecting “links”.
The One Mathematical/Abstract Aspect that is:
indispensable in studying discrete mathematical structures,
linked to a plethora of disciplines in a variety of fundamental fields,
behind the modelling of many, many real-world situations,
used for developing many computer scientific applications,
easily representable and analysable by computers,
both extremely classical and very modern, and
. . . more!
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 8 (of 25)
9. Computer Science Division: CS304
Basics of Graph Theory:
(Finite) Graphs: a formal definition
1 A graph G is a pair G = ⟨V , E⟩ consisting of:
1 a finite set V ̸= ∅, called the vertex set of G, and
2 a set E of two-element subsets of V called the edge set of G.
2 v ∈ V is called a node (or vertex).
3 e = {u, v} ∈ E is an edge (u ̸= v), with end vertices u and v.
4 For an edge e = {u, v} ∈ E:
the end nodes u and v are incident with the edge e in G.
the end nodes u and v are adjacent in G.
We may use e = uv (short for e = {u, v}) if the context is clear.
5 The notations V (G) and E(G) are used to denote the vertex set
and the edge set of a graph G, respectively.
Graph Drawings:
Depictions (visualisations) of graphs using curves and points (in Rm
).
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 9 (of 25)
10. Computer Science Division: CS304
Drawings:
Let G = ⟨V , E⟩ be a graph, such that:3
V = {p, q, r, s, t, u}, and
E = {{p, q}, {p, s}, {q, r}, {q, t}, {s, u}}.
Here is one drawing of G:
p
q
r
s
t
u
Here is another (identical):
p
q
r
s
t
u
. . . and there can be many other (isomorphic) drawings for G!
3It is important to note that G can be directly given in the form:
G = ⟨{p, q, r, s, t, u}, {{p, q}, {p, s}, {q, r}, {q, t}, {s, u}}⟩.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 10 (of 25)
11. Computer Science Division: CS304
(Concrete) Examples:
Shipping routes: Vertices are shipping hubs
Edges are routes between hubs.
Social networks: Vertices are people
Edges are social connections.
(e.g., e = {u, v} means u is a “friend” of v).
Telecommunications networks: Vertices are computers (on a network)
Edges are network connections between computers.
Disease transmission: Vertices are organisms (that carry the disease)
Edges represent one organism spreading it to another.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 11 (of 25)
12. Computer Science Division: CS304
More (and yet More) Basics: Pseudographs
In a Multi-set . . .
elements can occur more than once, and each element has a multiplicity
to give the number of times it appears in the multi-set.
(Multiple) Parallel Edges and Multi-graphs:
Edges that have the same end vertices are parallel.
A graph is called a multi-graph if its edge set is a multi-set.
(i.e., if its edge set has parallel edges connecting the same 2 nodes)
Loops:
A loop is a self-connecting edge.
(That’s, an edge connecting a node to itself in the form {v, v}.)
Loop Graphs:
A loop graph is a graph with at least one loop.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 12 (of 25)
13. Computer Science Division: CS304
More (and yet More) Basics:
Example:
A Graph G = ⟨V , E⟩ is called:
Null: if it has no vertices (V = ∅).
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 13 (of 25)
14. Computer Science Division: CS304
More (and yet More) Basics:
Example:
q
A Graph G = ⟨V , E⟩ is called:
Null: if it has no vertices (V = ∅).
Trivial: if it has only one node (|V | = 1).
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 13 (of 25)
15. Computer Science Division: CS304
More (and yet More) Basics:
Example:
p
q
r
s
t
u
A Graph G = ⟨V , E⟩ is called:
Null: if it has no vertices (V = ∅).
Trivial: if it has only one node (|V | = 1).
Empty: if it has no edges (E = ∅).
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 13 (of 25)
16. Computer Science Division: CS304
More (and yet More) Basics:
Example:
p
q
r
s
t
u
A Graph G = ⟨V , E⟩ is called:
Null: if it has no vertices (V = ∅).
Trivial: if it has only one node (|V | = 1).
Empty: if it has no edges (E = ∅).
Simple: if it has NEITHER a parallel edge NOR a loop.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 13 (of 25)
17. Computer Science Division: CS304
More (and yet More) Basics:
Example:
p
q
r
s
t
u
A Graph G = ⟨V , E⟩ is called:
Null: if it has no vertices (V = ∅).
Trivial: if it has only one node (|V | = 1).
Empty: if it has no edges (E = ∅).
Simple: if it has NEITHER a parallel edge NOR a loop.
Pseudograph: if it has BOTH parallel edges AND a loop.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 13 (of 25)
18. Computer Science Division: CS304
More (and yet More) Basics:
Vertex Degree:
The degree of a vertex counts the number of edges incident with it.
N.B. degG (v) denotes the degree of v ∈ V (G), for G = ⟨V , E⟩.
An isolated vertex v is a vertex with deg(v) = 0.
A pendant vertex v is a vertex with deg(v) = 1.
A loop is counted twice and parallel edges contribute separately.
Example:
p
q
rs
t
u
deg(u) = 1 (us is a pendant edge).
deg(p) = 2 (= deg(r) = deg(s)).
deg(t) = 3 (ACHTUNG!).
deg(q) = 4.
v∈V
deg(v) = 14 = 2 × |E|.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 14 (of 25)
19. Computer Science Division: CS304
More (and yet More) Basics:
(p, q)-Graphs:
If G = ⟨V , E⟩ with |V | = p and |E| = q, the G is called a (p, q)-graph.
Order & Size:
The number p is called the order of G, and the number q is called the
size of G.
The symbols p(G) and q(G) denotes the order and the size of G,
respectively.
Regular Graphs:
A graph G is regular of degree n, if all its nodes have the same degree n.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 15 (of 25)
20. Computer Science Division: CS304
Representations for Computer Programs
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 16 (of 25)
21. Computer Science Division: CS304
Graphs and Matrices: Part 0 – “The Adjacency Graph”
Assume that G = ⟨V , E⟩ is a graph with n = |V | vertices.
W.L.O.G., assume further that V (G) can be written as V = {vi }n
i=1.
The adjacency graph of G is the n × n square matrix [aij ]1≤i,j≤n
aij =
1 if {vi , vj } ∈ E(G)
0 if {vi , vj } ̸∈ E(G)
Example:
v1
v2
v3
v4
v5
v6
=⇒
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 1 0 1 0 0
1 0 1 0 1 0
0 1 0 0 0 0
1 0 0 0 0 1
0 1 0 0 0 0
0 0 0 1 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 17 (of 25)
22. Computer Science Division: CS304
Special (Simple) Graphs
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 18 (of 25)
23. Computer Science Division: CS304
Special Simple Graphs: Cycle Graphs
A simple graph G = ⟨V , E⟩ is a cycle if V can be written as
{v1, v2, · · · , vn}, and E = {{v1, v2} , {v2, v3} , · · · , {vn−1, vn} , {vn, v1}},
where n ≥ 3.
The cycle graph with n ≥ 3 nodes is denoted by Cn.
Examples:
Questions:
1 What is the degree of each vertex in Cn?
2 How many edges are there in Cn?
3 How does Cn’s adjacency matrix look like?
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 19 (of 25)
24. Computer Science Division: CS304
Special Simple Graphs: Path Graphs
A simple graph G = ⟨V , E⟩ is a path if V can be written as
{v1, v2, · · · , vn}, and E = {{v1, v2} , {v2, v3} , · · · , {vn−1, vn}}, where
n ≥ 2 (it can be drawn so that all vertices and edges lie on a single straight line).
The path graph with n ≥ 2 nodes is denoted by Pn.
Examples: Pn consists of 2 pendant nodes & n − 2 nodes of degree 2 .
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 20 (of 25)
25. Computer Science Division: CS304
Special Simple Graphs: Complete Graphs
A simple graph G = ⟨V , E⟩ is complete if it contains every possible edge
between all the vertices.
The complete graph with n ≥ 1 nodes is denoted by Kn.
Example:
K5
Questions:
1 What is the degree of each vertex in Kn?
2 How many edges are there in Kn?
3 How does Kn’s adjacency matrix look like?
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 21 (of 25)
26. Computer Science Division: CS304
Special Simple Graphs: Star Graphs
The star graph of order n is the complete bipartite graph K1,n.
The star graph of order n is denoted by Sn.
Example:
Here are the four star graphs, Si , for 3 ≤ i ≤ 6.
True or False?
Sn has exactly n pendant vertices, for n ≥ 1.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 22 (of 25)
27. Computer Science Division: CS304
Special Simple Graphs: Wheels
A wheel on n ≥ 3 vertices is a simple graph with n + 1 vertices
v1, v2, . . . , vn, vn+1, where the first n vertices form the cycle Cn and the
(n + 1)st
vertex is connected by an edge to each of the first n vertices.
The wheel graph on n ≥ 3 nodes has a total of n + 1 nodes and is
denoted by Wn.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 23 (of 25)
28. Computer Science Division: CS304
Assignment #1: (deadline: 19-February-2015, 11:59:59pm CLT)
Answer ALL the following:
(a)-Identify 2 Mistakes (according to the current lecture)
State why the following is NOT a well-defined graph G = ⟨E, V ⟩:
nodes: E = {m ∈ N − {0}}. (i.e., the set of all positive integers)
edges: V = {⟨m1, m2⟩ : m1 is odd and m2 is even}.
(i.e., two nodes are adjacent if the first is odd and the second is even)
(b)-Understanding Representations:
Let G = ⟨V , E⟩ be a graph given by the adjacency matrix [aij ]1≤i,j≤5,
where 1 ≤ i, j ≤ 5 and:
aij =
1 if i ̸= j
0 otherwise.
Give V (G), E(G), and a drawing of G.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 24 (of 25)
29. Computer Science Division: CS304
References:
Bender, E. A. and Williamson, S. G. (2013).
Foundations of Combinatorics with Applications.
Dover Books on Mathematics. Dover Publications.
Rosen, K. (2012).
Discrete Mathematics and Its Applications (7th edition).
McGraw-Hill Education.
Tucker, A. (2012).
Applied Combinatorics, 6th Edition.
Wiley Global Education.
Dr. Ahmed Ashry “Algorithmic Combinatorics 2 ” for 3rd Year Students — Lecture #1 25 (of 25)