The Clebsch Graph
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
Investigations on Unit Distance Property of Clebsch Graph and Its Complement
Proceedings of the World Congress on Engineering 2012 Vol I WCE 2012, July 4 - 6, 2012, London, U.K. Investigations on Unit Distance Property of Clebsch Graph and Its Complement Pratima Panigrahi and Uma kant Sahoo ∗yz Abstract|An n-dimensional unit distance graph is on parameters (16,5,0,2) and (16,10,6,6) respectively. a simple graph which can be drawn on n-dimensional These graphs are known to be unique in the respective n Euclidean space R so that its vertices are represented parameters [2]. In this paper we give unit distance n by distinct points in R and edges are represented by representation of Clebsch graph in the 3-dimensional Eu- closed line segments of unit length. In this paper we clidean space R3. Also we show that the complement of show that the Clebsch graph is 3-dimensional unit Clebsch graph is not a 3-dimensional unit distance graph. distance graph, but its complement is not. Keywords: unit distance graph, strongly regular The Petersen graph is the strongly regular graph on graphs, Clebsch graph, Petersen graph. parameters (10,3,0,1). This graph is also unique in its parameter set. It is known that Petersen graph is 2-dimensional unit distance graph (see [1],[3],[10]). It is In this article we consider only simple graphs, i.e. also known that Petersen graph is a subgraph of Clebsch undirected, loop free and with no multiple edges. The graph, see [[5], section 10.6]. study of dimension of graphs was initiated by Erdos et.al [3]. -
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. -
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana To cite this version: Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana. The Veldkamp Space of GQ(2,4). International Journal of Geometric Methods in Modern Physics, World Scientific Publishing, 2010, pp.1133-1145. 10.1142/S0219887810004762. hal-00365656v2 HAL Id: hal-00365656 https://hal.archives-ouvertes.fr/hal-00365656v2 Submitted on 6 Jul 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Veldkamp Space of GQ(2,4) M. Saniga,1 R. M. Green,2 P. L´evay,3 P. Pracna4 and P. Vrana3 1Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´aLomnica, Slovak Republic ([email protected]) 2Department of Mathematics, University of Colorado Campus Box 395, Boulder CO 80309-0395, U. S. A. ([email protected]) 3Department of Theoretical Physics, Institute of Physics Budapest University of Technology and Economics, H-1521 Budapest, Hungary ([email protected] and [email protected]) and 4J. Heyrovsk´yInstitute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejˇskova 3, CZ-182 23 Prague 8, Czech Republic ([email protected]) (6 July 2009) Abstract It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomor- phic to PG(5,2). -
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. -
The Merris Index of a Graph∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 10, pp. 212-222, August 2003 ELA THE MERRIS INDEX OF A GRAPH∗ FELIX GOLDBERG† AND GREGORY SHAPIRO† Abstract. In this paper the sharpness of an upper bound, due to Merris, on the independence number of a graph is investigated. Graphs that attain this bound are called Merris graphs.Some families of Merris graphs are found, including Kneser graphs K(v, 2) and non-singular regular bipar- tite graphs. For example, the Petersen graph and the Clebsch graph turn out to be Merris graphs. Some sufficient conditions for non-Merrisness are studied in the paper. In particular it is shown that the only Merris graphs among the joins are the stars. It is also proved that every graph is isomorphic to an induced subgraph of a Merris graph and conjectured that almost all graphs are not Merris graphs. Key words. Laplacian matrix, Laplacian eigenvalues, Merris index, Merris graph, Independence number. AMS subject classifications. 05C50, 05C69, 15A42. 1. Introduction. Let G be a finite simple graph. The Laplacian matrix L(G) (which we often write as simply L) is defined as the difference D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. It is not hard to show that L is a singular M-matrix and that it is positive semidefinite. The eigenvalues of L will be denoted by 0 = λ1 ≤ λ2 ≤ ...≤ λn. It is easy to show that λ2 =0ifandonlyifG is not connected. -
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). -
Distance-Regular Graphs with Strongly Regular Subconstituents
P1: KCU/JVE P2: MVG/SRK QC: MVG Journal of Algebraic Combinatorics KL434-03-Kasikova April 24, 1997 13:11 Journal of Algebraic Combinatorics 6 (1997), 247–252 c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands. Distance-Regular Graphs with Strongly Regular Subconstituents ANNA KASIKOVA Department of Mathematics, Kansas State University, Manhattan, KS 66506 Received December 3, 1993; Revised October 5, 1995 Abstract. In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a1 0 and ai 0,1 for i 2,...,d. = ∈{ } = Keywords: distance-regular graph, strongly regular graph, association scheme 1. Introduction Let 0 be a connected graph without loops and multiple edges, d d(0) be the diameter = of 0, V (0) be the set of vertices and v V(0) .Fori 1,...,d let 0i (u) be the set of =| | = vertices at distance i from u (‘subconstituent’) and ki 0i(u). We use the same notation =| | 0i (u) for the subgraph of 0 induced by the vertices in 0i (u). Distance between vertices u and v in 0 will be denoted by ∂(u,v). Recall that a connected graph is said to be distance regular if it is regular and for each i = 1,...,dthe numbers ai 0i(u) 01(v) , bi 0i 1(u) 01(v) , ci 0i 1(u) 01(v) =| ∩ | =| + ∩ | =| ∩ | are independent of the particular choice of u and v with v 0i (u). It is well known that in l ∈ this case all numbers p 0i(u) 0j(v) do not depend on the choice of the pair u, v i, j =| ∩ | with v 0l (u). -
Some Topics Concerning Graphs, Signed Graphs and Matroids
SOME TOPICS CONCERNING GRAPHS, SIGNED GRAPHS AND MATROIDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Vaidyanathan Sivaraman, M.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Prof. Neil Robertson, Advisor Prof. Akos´ Seress Prof. Matthew Kahle ABSTRACT We discuss well-quasi-ordering in graphs and signed graphs, giving two short proofs of the bounded case of S. B. Rao's conjecture. We give a characterization of graphs whose bicircular matroids are signed-graphic, thus generalizing a theorem of Matthews from the 1970s. We prove a recent conjecture of Zaslavsky on the equality of frus- tration number and frustration index in a certain class of signed graphs. We prove that there are exactly seven signed Heawood graphs, up to switching isomorphism. We present a computational approach to an interesting conjecture of D. J. A. Welsh on the number of bases of matroids. We then move on to study the frame matroids of signed graphs, giving explicit signed-graphic representations of certain families of matroids. We also discuss the cycle, bicircular and even-cycle matroid of a graph and characterize matroids arising as two different such structures. We study graphs in which any two vertices have the same number of common neighbors, giving a quick proof of Shrikhande's theorem. We provide a solution to a problem of E. W. Dijkstra. Also, we discuss the flexibility of graphs on the projective plane. We conclude by men- tioning partial progress towards characterizing signed graphs whose frame matroids are transversal, and some miscellaneous results. -
Lattices from Group Frames and Vertex Transitive Graphs
Frames Lattices Graphs Examples Lattices from group frames and vertex transitive graphs Lenny Fukshansky Claremont McKenna College (joint work with Deanna Needell, Josiah Park and Jessie Xin) Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: k Let f P R be a nonzero vector, then Gf “ tUf : U P Gu k is called a group frame (or G-frame). If G acts irreducibly on R , Gf is called an irreducible group frame. Irreducible group frames are always tight. Frames Lattices Graphs Examples Group frames Let G be a finite subgroup of Ok pRq, the k-dimensional real orthog- k onal group, then G acts on R by left matrix multiplication. -
Strongly Regular Graphs, Part 1 23.1 Introduction 23.2 Definitions
Spectral Graph Theory Lecture 23 Strongly Regular Graphs, part 1 Daniel A. Spielman November 18, 2009 23.1 Introduction In this and the next lecture, I will discuss strongly regular graphs. Strongly regular graphs are extremal in many ways. For example, their adjacency matrices have only three distinct eigenvalues. If you are going to understand spectral graph theory, you must have these in mind. In many ways, strongly-regular graphs can be thought of as the high-degree analogs of expander graphs. However, they are much easier to construct. Many times someone has asked me for a matrix of 0s and 1s that \looked random", and strongly regular graphs provided a resonable answer. Warning: I will use the letters that are standard when discussing strongly regular graphs. So λ and µ will not be eigenvalues in this lecture. 23.2 Definitions Formally, a graph G is strongly regular if 1. it is k-regular, for some integer k; 2. there exists an integer λ such that for every pair of vertices x and y that are neighbors in G, there are λ vertices z that are neighbors of both x and y; 3. there exists an integer µ such that for every pair of vertices x and y that are not neighbors in G, there are µ vertices z that are neighbors of both x and y. These conditions are very strong, and it might not be obvious that there are any non-trivial graphs that satisfy these conditions. Of course, the complete graph and disjoint unions of complete graphs satisfy these conditions. -
Integral Circulant Graphsଁ Wasin So
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 306 (2005) 153–158 www.elsevier.com/locate/disc Note Integral circulant graphsଁ Wasin So Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA Received 10 September 2003; received in revised form 6 September 2005; accepted 17 November 2005 Available online 4 January 2006 Abstract − In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2(n) 1 non-isomorphic integral circulant graphs on n vertices, where (n) is the number of divisors of n. © 2005 Elsevier B.V. All rights reserved. Keywords: Integral graph; Circulant graph; Graph spectrum 1. Graph spectra Let G be a simple graph which is a graph without loop, multi-edge, and orientation. Denote A(G) the adjacency matrix of G with respect to a given labeling of its vertices. Note that A(G) is a symmetric (0, 1)-matrix with zero diagonal entries. Adjacency matrices with respect to different labeling are permutationally similar. The spectrum of G, sp(G), is defined as the spectrum of A(G), which is the set of all eigenvalues of A(G) including multiplicity. Since the spectrum of a matrix is invariant under permutation similarity, sp(G) is independent of the labeling used in A(G). Because A(G) is a real symmetric matrix, sp(G) contains real eigenvalues only. Moreover, A(G) is an integral matrix and so its characteristic polynomial is monic and integral, hence its rational roots are integers. -
The Extendability of Matchings in Strongly Regular Graphs
The extendability of matchings in strongly regular graphs Sebastian M. Cioab˘a∗ Weiqiang Li Department of Mathematical Sciences Department of Mathematical Sciences University of Delaware University of Delaware Newark, DE 19707-2553, U.S.A. Newark, DE 19707-2553, U.S.A. [email protected] [email protected] Submitted: Feb 25, 2014; Accepted: Apr 29, 2014; Published: May 13, 2014 Mathematics Subject Classifications: 05E30, 05C50, 05C70 Abstract A graph G of even order v is called t-extendable if it contains a perfect matching, t < v=2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not 3-extendable. We also show that strongly regular graphs of valency k > 3 with λ > 1 are bk=3c-extendable k+1 (when µ 6 k=2) and d 4 e-extendable (when µ > k=2), where λ is the number of common neighbors of any two adjacent vertices and µ is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best pos- sible as there are strongly regular graphs of valency k that are not dk=2e-extendable. We show that the extendability of many strongly regular graphs of valency k is at least dk=2e − 1 and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order.