This document discusses radio labeling of grid graphs. It defines key terms like radio labeling, radio number, distance between vertices, and diameter. It summarizes recent findings about analyzing the fundamental characteristics of grids to determine bounds on the radio number. New tools have allowed establishing important results about radio numbers of grids and completely determining the radio number for some grids. The document outlines approaches for constructing upper and lower bounds on the radio number for grids and establishing conditions where the bounds meet.
This presentation is about electromagnetic fields, history of this theory and personalities contributing to this theory. Applications of electromagnetism. Vector Analysis and coordinate systems.
This presentation is about electromagnetic fields, history of this theory and personalities contributing to this theory. Applications of electromagnetism. Vector Analysis and coordinate systems.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
Study and Analysis of Multidimensional Hilbert Space Filling Curve and Its Ap...ijcseit
A map has to be designed to show the directions and the objects present in a specific area. So, it is
necessary to visit each and every points of that area. For that a Space Filling Curve can be used. SFC can
visit all the points present in a multi-dimensional data base. A spatial query can select geographical
features based on location or spatial relationship, and a Nearest Neighbor search can be used to find the
nearest object of a query object.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
Study and Analysis of Multidimensional Hilbert Space Filling Curve and Its Ap...ijcseit
A map has to be designed to show the directions and the objects present in a specific area. So, it is
necessary to visit each and every points of that area. For that a Space Filling Curve can be used. SFC can
visit all the points present in a multi-dimensional data base. A spatial query can select geographical
features based on location or spatial relationship, and a Nearest Neighbor search can be used to find the
nearest object of a query object.
In many scientific areas, systems can be described as interaction networks where elements correspond to vertices and interactions to edges. A variety of problems in those fields can deal with network comparison and characterization.
The problem of comparing and characterizing networks is the task of measuring their structural similarity and finding characteristics which capture structural information. In order to analyze complex networks, several methods can be combined, such as graph theory, information theory, and statistics.
In this project, we present methods for measuring Shannon’s entropy of graphs.
- Basic overview of transmission line analysis
-How transmission line analysis differs from basic circuit analysis
- How distributed circuit element differs from Lumped elements
-Links to be referred for Smith Chart
The presentation is about Adaptive Beamforming for high data-rate applications. Analog beamforming, which is considered a cost effective solution for consumer devices are investigated. Two adaptive analog beamforming algorithms, i.e., a well-known perturbation-based and dmr-based which overcomes the drawbacks of perturbation-based algorithm are discussed in-detail and their performance comparisons are made with the help of computer simulations. Also variation of single-port structure is considered and it's benefits are exploited with the help of modified analog beamforming algorithms.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
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.
Antenna radiation pattern is also called antenna pattern, far-field pattern. The antenna gain cannot be obtained from the radiation pattern, but the directivity coefficient is obtained from the radiation pattern. Antenna gain = directivity factor * antenna efficiency. Therefore, it is certain that the directional coefficient is greater than the gain.
The antenna gain is mainly manifested through the test of the radiation pattern. There are many kinds of test systems for testing the pattern. That is the microwave chamber. And the result of the test in the darkroom is only a result of comparison with the ideal symmetrical vibrator. It is known that the gain of an ideal symmetrical oscillator is 2.15dB. In this way, the gain of the antenna can be calculated according to the test level.
G=D*N%.
In general, the efficiency of the antenna is not 100%, so G<d. When calculating the directional coefficient D of the antenna, it is usually calculated based on the lobe width of the main lobe shown on the directional pattern, such as the half-power lobe width, that is, the lobe width at which the level drops by 3dB.
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.
1. Let 𝑑 𝑢, 𝑣 denote the distance between two vertices u and v on a graph G and let 𝑑𝑖𝑎𝑚(𝑔) denote the diameter of
such a graph 𝐺. A connected graph 𝐺 has a radio labeling 𝑓 if for all distinct pairs of vertices 𝑢 and 𝑣 of 𝐺,
𝒅 𝒖, 𝒗 + |𝒇 𝒖 − 𝒇 𝒗 ≥ 𝒅𝒊𝒂𝒎 𝒈 + 𝟏.
Using algebraic and combinatorial techniques, bounds for the radio number of 𝑃𝑛 □ 𝑃𝑛 can be established. By
analyzing the fundamental characteristics of abstract grids, tools may be developed to state important
findings about the radio number of grids and, ultimately, completely determine the radio number of the grids.
A graph 𝐺 is comprised of two main elements: 𝑽, a set of vertices, and 𝑬, a set of edges. Each edge is also defined
by an unordered pair of vertices that the edge connects. For example, in the path graph 𝑃5belows, the vertices are
defined from left-to-right as 𝑣1 through 𝑣5, and edge 𝑒2 connects vertices 𝑣2 and 𝑣3.
The distance 𝑑(𝑣𝑖, 𝑣𝑗) between two vertices 𝑣𝑖 and 𝑣𝑗 is the least number of edges connecting 𝑣𝑖 and 𝑣𝑗. In 𝑃5
above, the shortest path between 𝑣2 and 𝑣5 consists of 3 edges, namely 𝑒2, 𝑒3, and 𝑒4. Thus, 𝑑 𝑣2, 𝑣5 = 5.
The diameter of a graph is the maximum distance between any two vertices of a graph. 𝑑𝑖𝑎𝑚 𝑃5 = 4 because if
we take the distances between all pairs of, then we see that the greatest distance of 4 lies between 𝑣1 and 𝑣5.
A labeling of a graph is an assignment of labels to vertices. Vertex labels are typically assigned integer values and
all labels in this paper can be assumed to be represented by integers. Furthermore, a labeling of a graph can be
defined by a function or mapping that assigns labels algorithmically. A radio labeling is defined as an assignment
of vertex labels such that all distinct pairs of vertices of a graph satisfy the radio condition (see abstract).
The span of a labeling is the difference between the highest and lowest labels.
The radio number of a graph 𝐺, 𝒓𝒏(𝑮), is then the minimum possible span across all radio labelings of 𝐺. In other
words, if all radio labelings of a graph are considered, the radio number is the lowest span that we can find for that
graph under any of those radio labelings.
• Define a naïve labeling 𝒇 accordingly: 𝑓 𝑥𝑖 = 𝑓 𝑥𝑖−1 + 𝑑𝑖𝑎𝑚 𝐺 + 1 − 𝑑(𝑥𝑖, 𝑥𝑖−1).
• Define a 1st-order pair of vertices as any consecutively labeled vertices. We can think of an
nth-order pair of vertices as any pair of vertices 𝑥𝑖 and 𝑥𝑖+𝑛 for all 𝑖 ≥ 0.
• The term “bump” refers to the increase in span of a labeling associated with the
occurrence of some deviation from a purely distance-maximizing labeling scheme.
• An even grid can be subdivided into four quadrants, defined as 𝑄1, 𝑄2, 𝑄3, and 𝑄4 as
illustrated below. 𝑄1 and 𝑄3 are diagonally opposite quadrants. adjacent quadrants are
quadrants having the characteristic of not being diagonally opposite to one another – e.g., 𝑄1
and 𝑄2.
PROPOSITION 1 (diagonal triple ∝): Let 𝐺 be an even grid and 𝑥𝑖 a vertex labeled in any
quadrant of grid G. Say 𝑥𝑖 and 𝑥𝑖+1 are vertices in a diagonally opposite quadrant. Then for
any naïve labeling 𝑓 of 𝐺 and any radio labeling 𝑓∗
of 𝐺,
𝒇∗
𝒙𝒊+𝟏 − 𝒇∗
𝒙𝒊−𝟏 ≥ 𝒇 𝒙𝒊+𝟏 − 𝒇 𝒙𝒊−𝟏 + 𝟏,
for any 3 consecutively labeled vertices 𝑥𝑖−1, 𝑥𝑖, and𝑥𝑖+1.
LEMMA 1 (∝ independent triples): In an even grid 𝐺, given a distance-maximizing labeling
𝑐0 and any labeling 𝑐∗ that label vertices of 𝐺 in the same order,
𝒔𝒑𝒂𝒏 𝒄∗
≥ 𝒔𝒑𝒂𝒏 𝒄 𝟎 + ∝,
where ∝ is the number of diagonal triples that occur in the complete labeling of 𝐺 under 𝑐∗
.
LEMMA 2 (𝛽 adjacent flips): Given a distance-maximizing labeling 𝑐0 and a radio labeling
𝑐 of an even grid 𝐺,
𝒔𝒑𝒂𝒏 𝒄 ≥ 𝒔𝒑𝒂𝒏 𝒄 𝟎 + 𝟐 ∙ 𝜷,
where 𝛽 is the number of adjacent flips that occur in the complete labeling of 𝐺 under 𝑐∗
.
LEMMA 3 (∝ independent diagonal triples and 𝛽 adjacent flips): Let 𝐺 be an even grid
with distance-maximizing labeling 𝑐 and any other labeling 𝑐∗
. If 𝑐∗
contains ∝
independent diagonal triples and 𝛽 adjacent flips,
𝒔𝒑𝒂𝒏 𝒄∗
≥ 𝒔𝒑𝒂𝒏 𝒄 +∝ +𝟐 ∙ 𝜷.
In order to establish the upper bound for 𝑟𝑛 𝑃𝑛 □ 𝑃 𝑛 , we must:
• Specify the labeling order for vertices
• Label ordered vertices with minimum values that satisfy the radio condition
• Compute the span of this labeling upper bound of 𝑟𝑛 𝑃𝑛 □ 𝑃 𝑛
The span is computed by rearranging the radio condition and summing distances:
𝒇 𝒙𝒊+𝟏 ≤ 𝒅𝒊𝒂𝒎 𝑮 + 𝟏 − 𝒅(𝒙𝒊, 𝒙𝒊+𝟏) + 𝒇(𝒙𝒊)
We determine the minimal value for the last vertex labeled satisfying the radio condition…
…apply this to odd grids without loss of generality:
𝑓 𝑥(2𝑘+1)2 ≤ 𝑑𝑖𝑎𝑚 𝐺 + 1 − 𝑑(𝑥(2𝑘+1)2−1,𝑥(2𝑘+1)2) + 𝑓(𝑥(2𝑘+1)2−1)
𝑓 𝑥(2𝑘+1)2−1 ≤ 𝑑𝑖𝑎𝑚 𝐺 + 1 − 𝑑(𝑥(2𝑘+1)2−2,𝑥(2𝑘+1)2−1) + 𝑓(𝑥(2𝑘+1)2−2)
⋮
𝑓 𝑥1 ≥ 0
𝑠𝑝𝑎𝑛 𝑓 = 𝑓 𝑥(2𝑘+1)2 ≤
𝑗=1
𝑘
2𝑘 + 1 + 2𝑘(8𝑗 − 1)
THEOREM 1: 𝒓𝒏 𝑷 𝟐𝒌+𝟏 □ 𝑷 𝟐𝒌+𝟏 ≤ 𝟖𝒌 𝟑
+ 𝟖𝒌 𝟐
+ 𝒌
The lower bound for 𝑟𝑛 𝑃𝑛 □ 𝑃 𝑛 is established by maximizing distances.
Again, rearranging the radio condition and summing across all vertex pairs gives:
𝑓(𝑥𝑖+1) ≥ (𝑛2
− 1)(2𝑛 − 1)
𝑖=1
𝑛2−1
𝑑(𝑥𝑖+1, 𝑥𝑖)
• Let 𝝈 denote the row index and 𝝉 the column index of a vertex. Then,
• 𝒊=𝟏
𝒏 𝟐−𝟏
𝒅 𝒙𝒊+𝟏, 𝒙𝒊 = 𝝈 𝟏 − 𝝈 𝟐 + 𝝉 𝟏 − 𝝉 𝟐 + 𝝈 𝟐 − 𝝈 𝟑 + 𝝉 𝟐 − 𝝉 𝟑 + ⋯
⋯ + 𝝈 𝒏 𝟐−𝟏 − 𝝈 𝒏 𝟐 + 𝝉 𝒏 𝟐−𝟏 − 𝝉 𝒏 𝟐
• If we want to maximize distances, we must label vertices such that the highest row and column
indices are added in the sum, while the lowest are subtracted.
The first and last vertices labeled only contribute 2 indices each to the sum and
we choose arrangements of vertices that maximize the sum to minimize the span…
THEOREM 2.1: THEOREM 2.2:
𝒓𝒏 𝑷 𝟐𝒌+𝟏 □ 𝑷 𝟐𝒌+𝟏 ≥ 𝟖𝒌 𝟑
+ 𝟖𝒌 𝟐
− 𝟏 𝒓𝒏 𝑷 𝟐𝒌 □ 𝑷 𝟐𝒌 ≥ 𝟖𝒌 𝟑
− 𝟒𝒌 𝟐
− 𝟒𝒌 + 𝟐
[1] Calles, Eduardo and Gomez, Henry, via personal communication and electronic files from C. Wyels
[2] G. Chartrand and Erwin, David and Zhang, Ping and Harary, Frank. Radio labelings of graphs, Bull Inst.
Combin. Appl., 33 (2001), pp. 77-85.
[3] G. Chartrand, D. Erwin, and P. Zhang. A graph labeling problem suggested by FM channel restrictions, Bull
Inst. Combin. Appl., 43 (2005), pp. 43-57.
[4] Liu, Daphne Der-Fen and Zhu, Xuding. Multi-level distance labelings for paths and cycles, Siam J. Discrete
Math., 19 (1993) pp. 610-621.
Allow me to offer my humble gratitude to my wonderful advisor and professor DR.
CYNTHIA WYELS. In addition, I would also like to thank DR. IVONA GRZEGORCZYK,
as well as CSUCI MATHEMATICS for providing such a positive learning environment!
RADIO LABELING SQUARE GRIDS
Dev Ananda Advisor : Dr. Cynthia Wyels Master’s Thesis Project
Path graphs represent the best model for radio labeling square grids because grids are just constructions of paths.
Below are radio labelings of path graph 𝑃4. The first is a radio labeling 𝒇 of 𝑃4.The second is a minimal or
optimal radio labeling 𝒇∗
of 𝑃4 under which the span of the given labeling is as low as possible under the radio
condition for all pairs of vertices.
A grid graph is the Cartesian product of two path graphs, and square grids are composed of
two identical paths. Odd grid 𝑷 𝟑 □ 𝑷 𝟑 is depicted below as a set of vertices and edges, and
subsequently, as a set of vertices in an array of boxes. The connected set of boxes represent
vertices, and two vertices are considered adjacent if they share an edge. An optimal radio
labeling 𝒇 𝒐𝒑𝒕𝒊𝒎𝒂𝒍 of the grid is shown using the box representation with label values associated
the corresponding vertex. Here we see that 𝑟𝑛 𝑷 𝟑 □ 𝑷 𝟑 = 𝟏𝟕.
Label Order4 7 2
9 5
6 3 8
3x3 ordering
14 23 7
25 1
20 11 3
4 16 21
6 18 13
10
19
5
15
24
2
12
17
22
8
9
5x5 ordering
k+1,1
k,1
k+1,k+1
1,2k+11,k
2k+1,2k+12k+1,1 2k+1,k 2k+1,k+1
1,1
• The first vertex labeled is 𝒌 + 𝟏, 𝒌 + 𝟏
• Cyclic labeling proceeds in the order: top
right, top left, bottom left, bottom right
• The distance between each pair of vertices in
the same cycle = 𝟐𝒌 + 𝟏
• The distance between pairs of vertices in
consecutive (different) cycles = 𝟐𝒌
Substituting inequalities while incorporating the distances that our labeling
order above mandates yields the following general span inequality:
1
Q1
Q3Q4
Q2
𝒙𝒊
𝒙𝒊+𝟏𝒙𝒊−𝟏
an abstract grid subdivided
into quadrants 𝑸𝟏 → 𝑸𝟒
quadrants of 𝑷 𝟔 □ 𝑷 𝟔
with a diagonal triple
ACKNOWLEDGEMENTS
ABSTRACT
RADIO LABELING PATHS & GRIDS
RELEVANT DEFINITIONS
NEW FINDINGS, THE BUMP, & CLOSING THE GAP
REFERENCES
CONSTRUCTING THE UPPER BOUND
CONSTRUCTING THE LOWER BOUND
Considerations for advantageous ways of analyzing grids, with regards to the radio number,
allows us to refine bounds and establish conditions in which upper and lower bounds meet.
MOTIVATIONS: THE CHANNEL ASSIGNMENT PROBLEM
Radio labeling, in graph theory, is motivated by a problem originating from the FCC's regulation of
FM radio stations. The FCC categorizes radio stations into classes. Stations are then assigned channels
based on certain factors, including antenna height and signal power. The distances between stations also
affect what channel is assigned to a radio station of a given station class. Radio stations assigned the same
must be some minimum distance apart in order to satisfy FFC regulations. Stations that are in closer
proximity to one another must be channels that are farther apart, after considering station class. This is
called the CHANNEL ASSIGNMENT PROBLEM. Graph theory abstracts this problem from radio
transmitter and tower organizational grids, and its solution is the RADIO NUMBER.