DOCSLIB.ORG

A Questionable Distance-Regular Graph

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Questionable Distance-Regular Graph A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the de¯nition of the intersection algebra. We investigate two examples of distance-regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We investigate if these intersection numbers are nonnegative and feasible. 1. Introduction In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the de¯nition of the intersection algebra. We investigate two examples of distance- regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We investigate if these intersection numbers are nonnegative and feasible. 2. Preliminaries In this section we ¯x notation and review the basic de¯nitions regarding graphs, distance- regularity graphs, and intersection numbers. For more information concerning graphs, see West [1] or Biggs [3]. 2.1 Graphs A graph ¡ consists of a ¯nite set V , whose elements are called vertices, another ¯nite set E, whose elements are called edges, and a subset I of V x E, called the incidence relation such that each edge is incident with exactly two vertices, and no pair of edges is incident with the same pair of vertices. Two distinct vertices are said to be adjacent if there exists an edge incident to them both. A walk is a sequence of vertices such that consecutive vertices in the sequence are adjacent. A path is a walk with distinct vertices. A graph is connected if for each pair of distinct vertices there exists a path containing them. The length of a walk with n vertices is n ¡ 1. The distance between vertices u and v, denoted by @(u; v), is the length of the shortest path containing the vertices. The diameter, d, of a graph is the maximum distance between any two vertices in V (¡). A graph automorphism is a permutation of the vertex set that preserves the adjacency relation. The group of automorphisms of a graph, ¡, is denoted by Aut(¡). A graph ¡ is vertex-transitive if for every u; v 2 V (¡), there exists an automorphism mapping u to 1 v. A connected graph ¡ is distance-transitive if for any vertices u; v; x; y 2 V (¡) such that @(u; v) = @(x; y) there exists an automorphism g 2 Aut(¡) that maps u to x and v to y. 2.2 Distance-regular graphs Let ¡ be a connected graph with diameter d. For each u 2 V (¡) and each integer i, de¯ne ¡i(u) := fv 2 V (¡) j @(u; v) = ig: If k is a nonnegative integer, we say ¡ is regular, with valency k, if for all u 2 V (¡), h j¡1(u)j = k. We say ¡ is distance-regular, with intersection numbers pij (0 · h; i; j · d), whenever for all integers h; i; j (0 · h; i; j · d) and all u; v 2 V (¡) with @(u; v) = h, h j¡i(u) \ ¡j(v)j = pij: (1) These intersection numbers must be nonnegative and satisfy a triangle inequality, in that h pij = 0 if one of h, i, j is greater than the sum of the other two. Necessary conditions such as these on the intersection numbers are often called feasibility conditions. Unless a set of parameters meets these criteria, it is impossible (infeasible) to have a distance-regular graph with these intersection numbers. For more information on feasibility conditions, see [2] For convenience, we de¯ne i ci := pi¡1;1 (1 · i · d); i bi := pi+1;1 (0 · i · d ¡ 1); i ai := pi1 (1 · i · d) and de¯ne c0 = 0 and bd = 0. Note that a0 = 0 and c1 = 1. Also, for ¡ regular with valency k, we have k = b0. We can also observe that ci + ai + bi = k (0 · i · d): (2) For any vertex v of a distance-regular graph ¡ with diameter d, de¯ne ki = j¡i(v)j for 1 · i · d. It follows then that there are ki¡1 vertices in ¡i¡1(v) and each of those vertices is adjacent to bi¡1 vertices in ¡i(v). Also, there ki vertices in ¡i(v) that are adjacent to ci vertices in ¡i¡1(v). So counting the number of edges incident with both ¡i¡1(v) and ¡i(v) is ki¡1bi¡1 = kici. Thus, kb1b2 : : : bi¡1 ki = (3) c2c3 : : : ci For the remainder of this paper, we will assume ¡ is distance-regular with diameter d ¸ 3. 2 3. The Adjacency Algebra, A Given a graph, ¡ with vertex set fv1; v2; :::; vng, the adjacency matrix of ¡ is the n £ n matrix, A, whose entry (A)ij is the number of edges in ¡ with endpoints fvi; vjg. We notice that by our de¯nition of graph, A is a symmetric 0¡1 matrix. It also follows then that if ¸ is an eigenvalue of A, then ¸ is real and that the multiplicity of ¸ as a root of det(¸I ¡ A) = 0 is equal to the dimension of the space generated by the eigenvector corresponding to ¸. We will usually refer to the eigenvalues of A as the eigenvalues of ¡. De¯nition 3.0.1 Let ¡ be a distance-regular graph with diameter d and adjacency matrix A. De¯ne the adjacency algebra of ¡ to be the algebra given by spanfA0; A; A2;:::g and denoted by A. Proposition 3.0.2 For any nonnegative integer k, the (i; j)th entry of Ak counts the number of vi; vj-walks of length k. Proof. This holds for k = 0 since A0 is the identity matrix, I. For k = 1, A1 = A is the adjacency matrix, which by de¯nition counts the number of walks of length 1 between any l two vertices. Suppose true for A for l < k. Observe that every walk of length k from vi to vj consists of a walk of length k ¡ 1 from vi to a vertex vl that is adjacent to vj. By the k¡1 induction hypothesis, (A )il is the number of walks of length k ¡ 1 with endpoints fvi; vlg for 0 · l < k. Thus k k¡1 (A )ij = (A A)ij Xn k¡1 = (A )il(A)lj l=1 k¡1 k¡1 k¡1 = (A )i1(A)1k + (A )i2(A)2k + ¢ ¢ ¢ + (A )in(A)nj which equals the number of walks of length k with endpoints fvi; vjg. ¤ Lemma 3.0.3 Let ¡ be a distance-regular graph with diameter d and adjacency algebra A. Then dim(A) ¸ d + 1. Proof. Pick any u; v 2 V (¡) and let @(u; v) = k. Then for all i < k the uv-entry of Ai will be zero since the shortest path connecting u to v is of length k and the uv-entry of Ak is nonzero. Since the diameter of ¡ is d, there exist vertices x; y 2 V (¡) such that @(x; y) = i for 0 · i · d. So for all 0 · i · d, Ai is not contained in the span of A0; ¢ ¢ ¢ ;Ai¡1. Thus, fA0; A; : : : ; Adg are linearly independent and dim A ¸ d + 1. ¤ Corollary 3.0.4 A connected graph with diameter d has at least d+1 distinct eigenvalues. Proof. Let ¡ be a connected graph with adjacency matrix, A. By Lemma 3.0.3, the dimen- sion of A is at least d + 1, thus the minimum polynomial of A has degree at least d + 1. It follows then that A has at least d + 1 distinct eigenvalues. ¤ 3 3.1 Bose-Mesner Algebra, M We next recall the Bose-Mesner algebra of ¡. For each integer i (0 · i · d), let Ai be the ith distance matrix, which is the n £ n matrix where each rs-entry is given by ½ 1 if @(vr; vs) = i; (Ai)rs = (vr; vs 2 V (¡)): 0 if @(vr; vs) 6= i We notice that A0 = I and that A1 is the adjacency matrix A of ¡. Also, A0 + A1 + ::: + Ad = J; t where J is the all-1 matrix of size n. It is also clear that Ai = Ai for (0 · i · d). Lemma 3.1.1 Let ¡ be a distance-regular graph with diameter d. For 1 · i; j · d, Xd h AiAj = pijAh: h=0 th Proof. Let ¡ be a distance-regular graph with diameter d and Ak the k distance matrix of ¡ for (0 · k · d). Pick any x; y 2 V (¡) and let t = @(x; y). Then X (AiAj)xy = (Ai)xz(Aj)zy z2V (¡) = jfz 2 V (¡) j @(x; z) = i and @(z; y) = jgj t = pij: Pd h On the other hand, the xy-entry of h=0 pijAh is equal to Xd h t pij(Ah)xy = pij(At)xy h=0 t = pij(1) t = pij; since (Ah)xy = 0 if h 6= t. ¤ Corollary 3.1.2 Let ¡ be a distance-regular graph. For 1 · i · d, AAi = bi¡1Ai¡1 + aiAi + ci+1Ai+1: Proof. By applying Lemma 3.1.1 and the de¯nition of bi; ai, and ci, the equation follows easily. ¤ It follows then that 1 Ai+1 = [AAj ¡ bj¡1Aj¡1 ¡ ajAj]; ci+1 4 which is a polyomial in A with degree j + 1. Now, the distance matrices are linearly independent (exactly one matrix has a nonzero entry in every position) and, by Lemma 3.1.1, closed under multiplication.
Load more

Recommended publications
  • Distance Regular Graphs Simply Explained
    Distance Regular Graphs Simply Explained Alexander Coulter Paauwe April 20, 2007 Copyright c 2007 Alexander Coulter Paauwe. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Adjacency Algebra and Distance Regular Graphs: In this paper, we will discover some interesting properties of a particular kind of graph, called distance regular graphs, using algebraic graph theory. We will begin by developing some definitions that will allow us to explore the relationship between powers of the adjacency matrix of a graph and its eigenvalues, and ultimately give us particular insight into the eigenvalues of distance regular graphs. Basis for the Polynomial of an Adjacency Matrix We all know that for a given graph G, the powers of its adjacency matrix, A, have as their entries the number of n-walks between two vertices. More succinctly, we know that [An]i,j is the number of n-walks between the two vertices vi and vj. We also know that for a graph with diameter d, the first d powers of A are all linearly independent. Now, let us think of the set of all polynomials of the adjacency matrix, A, for a graph G. We can think of any member of the set as a linear combination of powers of A.
    [Show full text]
  • Constructing an Infinite Family of Cubic 1-Regular Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2002) 23, 559–565 doi:10.1006/eujc.2002.0589 Available online at http://www.idealibrary.com on Constructing an Infinite Family of Cubic 1-Regular Graphs YQAN- UAN FENG† AND JIN HO KWAK A graph is 1-regular if its automorphism group acts regularly on the set of its arcs. Miller [J. Comb. Theory, B, 10 (1971), 163–182] constructed an infinite family of cubic 1-regular graphs of order 2p, where p ≥ 13 is a prime congruent to 1 modulo 3. Marusiˇ cˇ and Xu [J. Graph Theory, 25 (1997), 133– 138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al.[J. Aust. Math. Soc. A, 56 (1994), 391–402] constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller’s construction can be generalized to an infinite family of cubic 1-regular graphs of order 2n, where n ≥ 13 is odd such that 3 divides ϕ(n), the Euler function of n. In this paper, we construct an infinite family of cubic 1-regular graphs with order 8(k2 + k + 1)(k ≥ 2) as cyclic-coverings of the three-dimensional Hypercube. c 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION In this paper we consider an undirected finite connected graph without loops or multiple edges. For a graph G, we denote by V (G), E(G), A(G) and Aut(G) the vertex set, the edge set, the arc set and the automorphism group, respectively.
    [Show full text]
  • Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
    SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1.
    [Show full text]
  • Algebraic Graph Theory: Automorphism Groups and Cayley Graphs
    Algebraic Graph Theory: Automorphism Groups and Cayley graphs Glenna Toomey April 2014 1 Introduction An algebraic approach to graph theory can be useful in numerous ways. There is a relatively natural intersection between the fields of algebra and graph theory, specifically between group theory and graphs. Perhaps the most natural connection between group theory and graph theory lies in finding the automorphism group of a given graph. However, by studying the opposite connection, that is, finding a graph of a given group, we can define an extremely important family of vertex-transitive graphs. This paper explores the structure of these graphs and the ways in which we can use groups to explore their properties. 2 Algebraic Graph Theory: The Basics First, let us determine some terminology and examine a few basic elements of graphs. A graph, Γ, is simply a nonempty set of vertices, which we will denote V (Γ), and a set of edges, E(Γ), which consists of two-element subsets of V (Γ). If fu; vg 2 E(Γ), then we say that u and v are adjacent vertices. It is often helpful to view these graphs pictorially, letting the vertices in V (Γ) be nodes and the edges in E(Γ) be lines connecting these nodes. A digraph, D is a nonempty set of vertices, V (D) together with a set of ordered pairs, E(D) of distinct elements from V (D). Thus, given two vertices, u, v, in a digraph, u may be adjacent to v, but v is not necessarily adjacent to u. This relation is represented by arcs instead of basic edges.
    [Show full text]
  • How Symmetric Are Real-World Graphs? a Large-Scale Study
    S S symmetry Article How Symmetric Are Real-World Graphs? A Large-Scale Study Fabian Ball * and Andreas Geyer-Schulz Karlsruhe Institute of Technology, Institute of Information Systems and Marketing, Kaiserstr. 12, 76131 Karlsruhe, Germany; [email protected] * Correspondence: [email protected]; Tel.: +49-721-608-48404 Received: 22 December 2017; Accepted: 10 January 2018; Published: 16 January 2018 Abstract: The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.
    [Show full text]
  • Perfect Domination in Book Graph and Stacked Book Graph
    International Journal of Mathematics Trends and Technology (IJMTT) – Volume 56 Issue 7 – April 2018 Perfect Domination in Book Graph and Stacked Book Graph Kavitha B N#1, Indrani Pramod Kelkar#2, Rajanna K R#3 #1Assistant Professor, Department of Mathematics, Sri Venkateshwara College of Engineering, Bangalore, India *2 Professor, Department of Mathematics, Acharya Institute of Technology, Bangalore, India #3 Professor, Department of Mathematics, Acharya Institute of Technology, Bangalore, India Abstract: In this paper we prove that the perfect domination number of book graph and stacked book graph are same as the domination number as the dominating set satisfies the condition for perfect domination. Keywords: Domination, Cartesian product graph, Perfect Domination, Book Graph, Stacked Book Graph. I. INTRODUCTION By a graph G = (V, E) we mean a finite, undirected graph with neither loops nor multiple edges. The order and size of G are denoted by p and q respectively. For graph theoretic terminology we refer to Chartrand and Lesnaik[3]. Graphs have various special patterns like path, cycle, star, complete graph, bipartite graph, complete bipartite graph, regular graph, strongly regular graph etc. For the definitions of all such graphs we refer to Harry [7]. The study of Cross product of graph was initiated by Imrich [12]. For structure and recognition of Cross Product of graph we refer to Imrich [11]. In literature, the concept of domination in graphs was introduced by Claude Berge in 1958 and Oystein Ore in [1962] by [14]. For review of domination and its related parameters we refer to Acharya et.al. [1979] and Haynes et.al.
    [Show full text]
  • Strongly Regular Graphs
    Chapter 4 Strongly Regular Graphs 4.1 Parameters and Properties Recall that a (simple, undirected) graph is regular if all of its vertices have the same degree. This is a strong property for a graph to have, and it can be recognized easily from the adjacency matrix, since it means that all row sums are equal, and that1 is an eigenvector. If a graphG of ordern is regular of degreek, it means that kn must be even, since this is twice the number of edges inG. Ifk�n− 1 and kn is even, then there does exist a graph of ordern that is regular of degreek (showing that this is true is an exercise worth thinking about). Regularity is a strong property for a graph to have, and it implies a kind of symmetry, but there are examples of regular graphs that are not particularly “symmetric”, such as the disjoint union of two cycles of different lengths, or the connected example below. Various properties of graphs that are stronger than regularity can be considered, one of the most interesting of which is strong regularity. Definition 4.1.1. A graphG of ordern is called strongly regular with parameters(n,k,λ,µ) if every vertex ofG has degreek; • ifu andv are adjacent vertices ofG, then the number of common neighbours ofu andv isλ (every • edge belongs toλ triangles); ifu andv are non-adjacent vertices ofG, then the number of common neighbours ofu andv isµ; • 1 k<n−1 (so the complete graph and the null graph ofn vertices are not considered to be • � strongly regular).
    [Show full text]
  • On Cayley Graphs of Algebraic Structures
    On Cayley graphs of algebraic structures Didier Caucal1 1 CNRS, LIGM, University Paris-East, France [email protected] Abstract We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree. 1 Introduction To describe the structure of a group, Cayley introduced in 1878 [7] the concept of graph for any group (G, ·) according to any generating subset S. This is simply the set of labeled s oriented edges g −→ g·s for every g of G and s of S. Such a graph, called Cayley graph, is directed and labeled in S (or an encoding of S by symbols called letters or colors). The study of groups by their Cayley graphs is a main topic of algebraic graph theory [3, 8, 2]. A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [15] : an unlabeled and undirected graph is a Cayley graph if and only if we can find a group with a free and transitive action on the graph. However, this algebraic characterization is not well suited for deciding whether a possibly infinite graph is a Cayley graph. It is pertinent to look for characterizations by graph-theoretic conditions. This approach was clearly stated by Hamkins in 2010: Which graphs are Cayley graphs? [10]. In this paper, we present simple graph-theoretic characterizations of Cayley graphs for firstly left-cancellative and cancellative monoids, and then for groups. These characterizations are then extended to any subset S of left-cancellative magmas, left-quasigroups, quasigroups, and groups.
    [Show full text]
  • 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..
    [Show full text]
  • On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
    ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph.
    [Show full text]
  • 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 $G_{x}$ .
    [Show full text]
  • 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.
    [Show full text]