On the Generalized Θ-Number and Related Problems for Highly Symmetric Graphs
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Connectivities and Diameters of Circulant Graphs
CONNECTIVITIES AND DIAMETERS OF CIRCULANT GRAPHS Paul Theo Meijer B.Sc. (Honors), Simon Fraser University, 1987 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (MATHEMATICS) in the Department of Mathematics and Statistics @ Paul Theo Meijer 1991 SIMON FRASER UNIVERSITY December 1991 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. Approval Name: Paul Theo Meijer Degree: Master of Science (Mathematics) Title of Thesis: Connectivities and Diameters of Circulant Graphs Examining Committee: Chairman: Dr. Alistair Lachlan Dr. grian Alspach, Professor ' Senior Supervisor Dr. Luis Goddyn, Assistant Professor - ph Aters, Associate Professor . Dr. Tom Brown, Professor External Examiner Date Approved: December 4 P 1991 PART IAL COPYH IGIiT L ICLNSI: . , I hereby grant to Sirnori Fraser- llr~ivorsitytho righl to lend my thesis, project or extended essay (tho title of which is shown below) to users of the Simon Frasor University Libr~ry,and to make part ial or single copies only for such users or in response to a request from the library of any other university, or other educational insfitution, on 'its own behalf or for one of its users. I further agree that percnission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission. Title of Thesis/Project/Extended Essay (date) Abstract Let S = {al, az, . -
JMM 2017 Student Poster Session Abstract Book
Abstracts for the MAA Undergraduate Poster Session Atlanta, GA January 6, 2017 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Organized by the MAA Committee on Undergraduate Student Activities and Chapters and CUPM Subcommittee on Research by Undergraduates Dear Students, Advisors, Judges and Colleagues, If you look around today you will see over 300 posters and nearly 500 student presenters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and universities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new Ph.D.s, and administrators. We have acknowledged many of them in this booklet; however, many judges here volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by Tudor Investments and Two Sigma. We are also helped by the mem- bers of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Dora C. Ahmadi; Jennifer Bergner; Benjamin Galluzzo; Kristina Cole Garrett; TJ Hitchman; Cynthia Huffman; Aihua Li; Sara Louise Malec; Lisa Marano; May Mei; Stacy Ann Muir; Andy Nieder- maier; Pamela A. Richardson; Jennifer Schaefer; Peri Shereen; Eve Torrence; Violetta Vasilevska; Gerard A. -
Graph Theory, Prentice -Hall of India
Annexure 145 www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 3; August 2011 Product Cordial Labeling in the Context of Tensor Product of Graphs S K Vaidya (Corresponding author) Department of Mathematics, Saurashtra University Rajkot-360 005. GUJARAT, India E-mail: [email protected] N B Vyas Atmiya Institute of Technology and Science Rajkot-360 005. GUJARAT, India E-mail: [email protected] Received: March 10, 2011 Accepted: March 28, 2011 doi:10.5539/jmr.v3n3p83 Abstract For the graph G1 and G2 the tensor product is denoted by G1(T p)G2 which is the graph with vertex set V(G1(T p)G2) = V(G1) × V(G2) and edge set E(G1(T p)G2) = {(u1, v1), (u2, v2)/u1u2E(G1) and v1v2E(G2)}. The graph Pm(T p)Pn is disconnected for ∀m, n while the graphs Cm(T p)Cn and Cm(T p)Pn are disconnected for both m and n even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling. Keywords: Cordial labeling, Porduct cordial labeling, Tensor product AMS Subject classification (2010): 05C78. 1. Introduction We begin with simple, finite and undirected graph G = (V(G), E(G)). For standard terminology and notations we follow (West, D. B, 2001). The brief summary of definitions and relevant results are given below. 1.1 Definition: If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling. -
Graph Operations and Upper Bounds on Graph Homomorphism Counts
Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops). -
Some New General Lower Bounds for Mixed Metric Dimension of Graphs
Some new general lower bounds for mixed metric dimension of graphs Milica Milivojevi´cDanasa, Jozef Kraticab, Aleksandar Savi´cc, Zoran Lj. Maksimovi´cd, aFaculty of Science and Mathematics, University of Kragujevac, Radoja Domanovi´ca 12, Kragujevac, Serbia bMathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11 000 Belgrade, Serbia cFaculty of Mathematics, University of Belgrade, Studentski trg 16/IV, 11000 Belgrade, Serbia dUniversity of Defence, Military Academy, Generala Pavla Juriˇsi´ca Sturmaˇ 33, 11000 Belgrade, Serbia Abstract Let G = (V, E) be a connected simple graph. The distance d(u, v) between vertices u and v from V is the number of edges in the shortest u − v path. If e = uv ∈ E is an edge in G than distance d(w, e) where w is some vertex in G is defined as d(w, e) = min(d(w,u),d(w, v)). Now we can say that vertex w ∈ V resolves two elements x, y ∈ V ∪ E if d(w, x) =6 d(w,y). The mixed resolving set is a set of vertices S, S ⊆ V if and only if any two elements of E ∪ V are resolved by some element of S. A minimal resolving set related to inclusion is called mixed resolving basis, and its cardinality is called the mixed metric dimension of a graph G. This graph invariant is recently introduced and it is of interest to find its general properties and determine its values for various classes of graphs. Since arXiv:2007.05808v1 [math.CO] 11 Jul 2020 the problem of finding mixed metric dimension is a minimization problem, of interest is also to find lower bounds of good quality. -
Distances, Diameter, Girth, and Odd Girth in Generalized Johnson Graphs
Distances, Diameter, Girth, and Odd Girth in Generalized Johnson Graphs Ari J. Herman Dept. of Mathematics & Statistics Portland State University, Portland, OR, USA Mathematical Literature and Problems project in partial fulfillment of requirements for the Masters of Science in Mathematics Under the direction of Dr. John Caughman with second reader Dr. Derek Garton Abstract Let v > k > i be non-negative integers. The generalized Johnson graph, J(v; k; i), is the graph whose vertices are the k-subsets of a v-set, where vertices A and B are adjacent whenever jA \ Bj = i. In this project, we present the results of the paper \On the girth and diameter of generalized Johnson graphs," by Agong, Amarra, Caughman, Herman, and Terada [1], along with a number of related additional results. In particular, we derive general formulas for the girth, diameter, and odd girth of J(v; k; i). Furthermore, we provide a formula for the distance between any two vertices A and B in terms of the cardinality of their intersection. We close with a number of possible future directions. 2 1. Introduction In this project, we present the results of the paper \On the girth and diameter of generalized Johnson graphs," by Agong, Amarra, Caughman, Herman, and Terada [1], along with a number of related additional results. Let v > k > i be non-negative integers. The generalized Johnson graph, X = J(v; k; i), is the graph whose vertices are the k-subsets of a v-set, with adjacency defined by A ∼ B , jA \ Bj = i: (1) Generalized Johnson graphs were introduced by Chen and Lih in [4]. -
ISOMETRIC SUBGRAPHS of HAMMING GRAPHS and D-CONVEXITY
4. K.V. Rudakov, "Completeness and universal constraints in the problem of correction of heuristic classification algorithms," Kibernetika, No, 3, 106-109 (1987). 5. K.V. Rudakov, "Symmetry and function constraints in the problem of correction of heur- istic classification algorithms," Kibernetika, No. 4, 74-77 (1987). 6. K.V. Rudakov, On Some Classes of Recognition Algorithms (General Results) [in Russian], VTs AN SSSR, Moscow (1980). ISOMETRIC SUBGRAPHS OF HAMMING GRAPHS AND d-CONVEXITY V. D. Chepoi UDC 519.176 In this study, we provide criteria of isometric embeddability of graphs in Hamming graphs. We consider ordinary connected graphs with a finite vertex set endowed with the natural metric d(x, y), equal to the number of edges in the shortest chain between the ver- tices x and y. Let al,...,a n be natural numbers. The Hamming graph Hal...a n is the graph with the ver- tex set X = {x = (x l..... xn):l~xi~a~, ~ = l,...n} in which two vertices are joined by an edge if and only if the corresponding vectors differ precisely in one coordinate [i, 2]. In other words, the graph Hal...a n is the Cartesian product of the graphs Hal,...,Han, where Hal is the ai-vertex complete graph. It is easy to show that in the Hamming graph the distance d(x, y) between the vertices x, y is equal to the number of different pairs of coordinates in the tuples corresponding to these vertices, i.e., it is equal to the Hamming distance between these tuples. It is also easy to show that the graph of the n-dimensional cube Qn may be treated as the Hamming graph H2.. -
Towards a Classification of Distance-Transitive Graphs
数理解析研究所講究録 1063 巻 1998 年 72-83 72 Towards a classification of distance-transitive graphs John van Bon Abstract We outline the programme of classifying all finite distance-transitive graphs. We men- tion the most important classification results obtained so far and give special attention to the so called affine graphs. 1. Introduction The graphs in this paper will be always assumed to be finite, connected, undirected and without loops or multiple edges. The edge set of a graph can thus be identified with a subset of the set of unoidered pairs of vertices. Let $\Gamma=(V\Gamma, E\Gamma)$ be a graph and $x,$ $y\in V\Gamma$ . With $d(x, y)$ we will denote the usual distance $\Gamma$ in between the vertices $x$ and $y$ (i.e., the length of the shortest path connecting $x$ and $y$ ) and with $d$ we will denote the diameter of $\Gamma$ , the maximum of all possible values of $d(x, y)$ . Let $\Gamma_{i}(x)=\{y|y\in V\Gamma, d(x, y)=i\}$ be the set of all vertices at distance $i$ of $x$ . An $aut_{omo}rph7,sm$ of a graph is a permutation of the vertex set that maps edges to edges. Let $G$ be a group acting on a graph $\Gamma$ (i.e. we are given a morphism $Garrow Aut(\Gamma)$ ). For a $x\in V\Gamma$ $x^{g}$ vertex and $g\in G$ the image of $x$ under $g$ will be denoted by . The set $\{g\in G|x^{g}=x\}$ is a subgroup of $G$ , called that stabilizer in $G$ of $x$ and will be denoted by $G_{x}$ . -
On J-Colorability of Certain Derived Graph Classes
On J-Colorability of Certain Derived Graph Classes Federico Fornasiero Department of mathemathic Universidade Federal de Pernambuco Recife, Pernambuco, Brasil [email protected] Sudev Naduvath Centre for Studies in Discrete Mathematics Vidya Academy of Science & Technology Thrissur, Kerala, India. [email protected] Abstract A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every colour class of G with respect to C. A maximal proper colouring of a graph G is a J-colouring of G if and only if every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-colouring of certain Mycielski type graphs. Key Words: Mycielski graphs, graph coloring, rainbow neighbourhoods, J-coloring arXiv:1708.09798v2 [math.GM] 4 Sep 2017 of graphs. Mathematics Subject Classification 2010: 05C15, 05C38, 05C75. 1 Introduction For general notations and concepts in graphs and digraphs we refer to [1, 3, 13]. For further definitions in the theory of graph colouring, see [2, 4]. Unless specified otherwise, all graphs mentioned in this paper are simple, connected and undirected graphs. 1 2 On J-colorability of certain derived graph classes 1.1 Mycielskian of Graphs Let G be a triangle-free graph with the vertex set V (G) = fv1; : : : ; vng. The Myciel- ski graph or the Mycielskian of a graph G, denoted by µ(G), is the graph with ver- tex set V (µ(G)) = fv1; v2; : : : ; vn; u1; u2; : : : ; un; wg such that vivj 2 E(µ(G)) () vivj 2 E(G), viuj 2 E(µ(G)) () vivj 2 E(G) and uiw 2 E(µ(G)) for all i = 1; : : : ; n. -
On the Wiener Polarity Index of Lattice Networks
RESEARCH ARTICLE On the Wiener Polarity Index of Lattice Networks Lin Chen1, Tao Li2*, Jinfeng Liu1, Yongtang Shi1, Hua Wang3 1 Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, China, 2 College of Computer and Control Engineering, Nankai University, Tianjin 300071, China, 3 Department of Mathematical Sciences Georgia Southern University, Statesboro, GA 30460-8093, United States of America * [email protected] a11111 Abstract Network structures are everywhere, including but not limited to applications in biological, physical and social sciences, information technology, and optimization. Network robustness is of crucial importance in all such applications. Research on this topic relies on finding a suitable measure and use this measure to quantify network robustness. A number of dis- OPEN ACCESS tance-based graph invariants, also known as topological indices, have recently been incor- Citation: Chen L, Li T, Liu J, Shi Y, Wang H (2016) porated as descriptors of complex networks. Among them the Wiener type indices are the On the Wiener Polarity Index of Lattice Networks. most well known and commonly used such descriptors. As one of the fundamental variants PLoS ONE 11(12): e0167075. doi:10.1371/journal. of the original Wiener index, the Wiener polarity index has been introduced for a long time pone.0167075 and known to be related to the cluster coefficient of networks. In this paper, we consider the Editor: Cheng-Yi Xia, Tianjin University of value of the Wiener polarity index of lattice networks, a common network structure known Technology, CHINA for its simplicity and symmetric structure. We first present a simple general formula for com- Received: August 25, 2016 puting the Wiener polarity index of any graph. -
On the Classification of Locally Hamming Distance-Regular Graphs
数理解析研究所講究録 第 76850巻 1991 年 50-61 On the Classification of Locally Hamming Distance-Regular Graphs BY MAKOTO MATSUMOTO Abstract. A distance-regular graph is locally Hamming if it is locally isomorphic to a Hamming scheme $H(r, 2)$ . This paper rediscovers the connection among locally Ham- ming distance-regular graphs, designs, and multiply transitive permutation groups, through which we classify some of locally Hamming distance-transitive graphs. \S 1. Introduction. By a graph we shall mean a finite undirected graph with no loops and no multiple edges. For a graph $G,$ $V(G)$ denotes the vertex set and $E(G)$ denotes the edge set of $G$ . For a vertex $v$ of a graph $G,$ $N(v)$ denotes the set of adjacent vertices with $v$ . By $\ovalbox{\tt\small REJECT}_{2}$ we denote the two-element field, and by $H(r)$ we denote the r-dimensional Hamming scheme for $r\geq 1$ ; that is, $H(r)$ is such a graph that its vertex set is the $\#_{2}^{=r}$ vector space a,nd $\tau\iota,$ $v\in\#_{2}^{=r}$ are adjacent if and only if the Hamming distance $d(u, v)=1$ ; i.e., $\#\{i u_{i}\neq v_{i}\}=1$ where $u=(u_{1}, \ldots u_{r})$ and $v=(v_{1}, \ldots , v_{r})$ . The graph obtained from $H(r)$ by identifying antipodal points is called a folded Hamming sheme(or a folded Hamming cube in [3, p.140]). The graph $H(3)$ is called a cube, and the induced subgraph obtained by removing one vertex tog$e$ ther with the three incident edges from a cube is called a tulip. -
Properties of Codes in the Johnson Scheme
Properties of Codes in the Johnson Scheme Natalia Silberstein Properties of Codes in the Johnson Scheme Research Thesis In Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics Natalia Silberstein Submitted to the Senate of the Technion - Israel Institute of Technology Shvat 5767 Haifa February 2007 The Research Thesis Was Done Under the Supervision of Professor Tuvi Etzion in the Faculty of Mathematics I wish to express my sincere gratitude to my supervisor, Prof. Tuvi Etzion for his guidance and kind support. I would also like to thank my family and my friends for thear support. The Generous Financial Help Of The Technion and Israeli Science Foundation Is Gratefully Acknowledged. Contents Abstract 1 List of symbols and abbreviations 3 1 Introduction 4 1.1 Definitions.................................. 5 1.1.1 Blockdesigns............................ 5 1.2 PerfectcodesintheHammingmetric. .. 7 1.3 Perfect codes in the Johnson metric (survey of known results)....... 8 1.4 Organizationofthiswork. 13 2 Perfect codes in J(n, w) 15 2.1 t-designs and codes in J(n, w) ....................... 15 2.1.1 Divisibility conditions for 1-perfect codes in J(n, w) ....... 17 2.1.2 Improvement of Roos’ bound for 1-perfect codes . 20 2.1.3 Number theory’s constraints for size of Φ1(n, w) ......... 24 2.2 Moments .................................. 25 2.2.1 Introduction............................. 25 2.2.1.1 Configurationdistribution . 25 2.2.1.2 Moments. ........................ 26 2.2.2 Binomial moments for 1-perfect codes in J(n, w) ........ 27 2.2.2.1 Applicationsof Binomial momentsfor 1-perfect codes in J(n, w) .......................