The Smith and Critical Groups of the Square Rook's Graph and Its
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
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). -
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. -
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. -
Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs
mathematics Article Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs Jianqiang Hao 1,*, Yunzhan Gong 2, Jianzhi Sun 1 and Li Tan 1 1 Beijing Key Laboratory of Big Data Technology for Food Safety, Beijing Technology and Business University, No. 11, Fu Cheng Road, Beijing 100048, China 2 State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, No 10, Xitucheng Road, Haidian District, Beijing 100876, China * Correspondence: [email protected]; Tel.: +86-10-6898-5704 Received: 5 July 2019; Accepted: 27 July 2019; Published: 31 July 2019 Abstract: This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to Nauty’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its open k-neighborhood subgraph . It defines dif fusion degree sequences and entire dif fusion degree sequence . For each node of a graph G, it designs a characteristic m_NearestNode to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of MaxQ(G). Another theorem presents how to determine the second nodes of MaxQ(G). When computing Cmax(G), if MaxQ(G) already holds the first i nodes u1, u2, ··· , ui, Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of MaxQ(G). Further, it also establishes two theorems to determine the Cmax(G) of disconnected graphs. -
Quantum Automorphism Groups of Finite Graphs
Quantum automorphism groups of finite graphs Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften der Fakult¨atf¨urMathematik und Informatik der Universit¨atdes Saarlandes Simon Schmidt Saarbr¨ucken, 2020 Datum der Verteidigung: 2. Juli 2020 Dekan: Prof. Dr. Thomas Schuster Pr¨ufungsausschuss: Vorsitzende: Prof. Dr. Gabriela Weitze-Schmith¨usen Berichterstatter: Prof. Dr. Moritz Weber Prof. Dr. Roland Speicher Prof. Dr. Julien Bichon Akademischer Mitarbeiter: Dr. Tobias Mai ii Abstract The present work contributes to the theory of quantum permutation groups. More specifically, we develop techniques for computing quantum automorphism groups of finite graphs and apply those to several examples. Amongst the results, we give a criterion on when a graph has quantum symmetry. By definition, a graph has quantum symmetry if its quantum automorphism group does not coincide with its classical automorphism group. We show that this is the case if the classical automorphism group contains a pair of disjoint automorphisms. Furthermore, we prove that several families of distance-transitive graphs do not have quantum symmetry. This includes the odd graphs, the Hamming graphs H(n; 3), the Johnson graphs J(n; 2), the Kneser graphs K(n; 2) and all cubic distance-transitive graphs of order ≥ 10. In particular, this implies that the Petersen graph does not have quantum symmetry, answering a question asked by Banica and Bichon in 2007. Moreover, we show that the Clebsch graph does have quantum symmetry and −1 prove that its quantum automorphism group is equal to SO5 answering a question asked by Banica, Bichon and Collins. More generally, for odd n, the quantum −1 automorphism group of the folded n-cube graph is SOn . -
Spectra of Graphs
Spectra of graphs Andries E. Brouwer Willem H. Haemers 2 Contents 1 Graph spectrum 11 1.1 Matricesassociatedtoagraph . 11 1.2 Thespectrumofagraph ...................... 12 1.2.1 Characteristicpolynomial . 13 1.3 Thespectrumofanundirectedgraph . 13 1.3.1 Regulargraphs ........................ 13 1.3.2 Complements ......................... 14 1.3.3 Walks ............................. 14 1.3.4 Diameter ........................... 14 1.3.5 Spanningtrees ........................ 15 1.3.6 Bipartitegraphs ....................... 16 1.3.7 Connectedness ........................ 16 1.4 Spectrumofsomegraphs . 17 1.4.1 Thecompletegraph . 17 1.4.2 Thecompletebipartitegraph . 17 1.4.3 Thecycle ........................... 18 1.4.4 Thepath ........................... 18 1.4.5 Linegraphs .......................... 18 1.4.6 Cartesianproducts . 19 1.4.7 Kronecker products and bipartite double. 19 1.4.8 Strongproducts ....................... 19 1.4.9 Cayleygraphs......................... 20 1.5 Decompositions............................ 20 1.5.1 Decomposing K10 intoPetersengraphs . 20 1.5.2 Decomposing Kn into complete bipartite graphs . 20 1.6 Automorphisms ........................... 21 1.7 Algebraicconnectivity . 22 1.8 Cospectralgraphs .......................... 22 1.8.1 The4-cube .......................... 23 1.8.2 Seidelswitching. 23 1.8.3 Godsil-McKayswitching. 24 1.8.4 Reconstruction ........................ 24 1.9 Verysmallgraphs .......................... 24 1.10 Exercises ............................... 25 3 4 CONTENTS 2 Linear algebra 29 2.1 -
Edge-Transitive Graphs on 20 Vertices Or Less
Edge-Transitive Graphs Heather A. Newman Hector Miranda Department of Mathematics School of Mathematical Sciences Princeton University Rochester Institute of Technology Darren A. Narayan School of Mathematical Sciences Rochester Institute of Technology September 15, 2017 Abstract A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all connected edge-transitive graphs on less than or equal to 20 vertices. We then present a construction for an infinite family of edge-transitive bipartite graphs, and use this construction to show that there exists a non-trivial bipartite subgraph of Km,n that is connected and edge-transitive whenever gcdpm, nq ą 2. Additionally, we investigate necessary and sufficient conditions for edge transitivity of connected pr, 2q biregular subgraphs of Km,n, as well as for uniqueness, and use these results to address the case of gcdpm, nq “ 2. We then present infinite families of edge-transitive graphs among vertex-transitive graphs, including several classes of circulant graphs. In particular, we present necessary conditions and sufficient conditions for edge-transitivity of certain circulant graphs. 1 Introduction A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on its vertex (edge) set. We note the alternative definition given by Andersen et al. [1]. Theorem 1.1 (Andersen, Ding, Sabidussi, and Vestergaard) A finite simple graph G is edge-transitive if and only if G ´ e1 – G ´ e2 for all pairs of edges e1 and e2. -
Strongly Regular Graphs
Strongly regular graphs Peter J. Cameron Queen Mary, University of London London E1 4NS U.K. Draft, April 2001 Abstract Strongly regular graphs form an important class of graphs which lie somewhere between the highly structured and the apparently random. This chapter gives an introduction to these graphs with pointers to more detailed surveys of particular topics. 1 An example Consider the Petersen graph: ZZ Z u Z Z Z P ¢B Z B PP ¢ uB ¢ uB Z ¢ B ¢ u Z¢ B B uZ u ¢ B ¢ Z B ¢ B ¢ ZB ¢ S B u uS ¢ B S¢ u u 1 Of course, this graph has far too many remarkable properties for even a brief survey here. (It is the subject of a book [17].) We focus on a few of its properties: it has ten vertices, valency 3, diameter 2, and girth 5. Of course these properties are not all independent. Simple counting arguments show that a trivalent graph with diameter 2 has at most ten vertices, with equality if and only it has girth 5; and, dually, a trivalent graph with girth 5 has at least ten vertices, with equality if and only it has diameter 2. The conditions \diameter 2 and girth 5" can be rewritten thus: two adjacent vertices have no common neighbours; two non-adjacent vertices have exactly one common neighbour. Replacing the particular numbers 10, 3, 0, 1 here by general parameters, we come to the definition of a strongly regular graph: Definition A strongly regular graph with parameters (n; k; λ, µ) (for short, a srg(n; k; λ, µ)) is a graph on n vertices which is regular with valency k and has the following properties: any two adjacent vertices have exactly λ common neighbours; • any two nonadjacent vertices have exactly µ common neighbours. -
Characterization of Line Graphs
INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of die original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of die page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. -
The Chang Graphs
Nadia Hamoudi The Chang graphs Abstract Chang showed that there are up to isomorphism four strongly regular graphs with parameters v = 28, k = 12, λ = 6, μ =4, namely T(8) and three other graphs, known as the Chang graphs. He proved that the these three graphs can be obtained by Seidel switching from T(8) (the line graph of K8), with respect to the edge set of 4 K2, K3 + K5 and C8. Introduction The work of Shrikhande (1959), A.J.Hoffman in the late 1950’s and Chang (1959, 1960) have contributed to the development of Seidel’s proof of the classification of strongly regular graphs with least eigenvalues -2. Seidel proved that strongly regular graphs with least eigenvalues -2 are: the triangular graphs, the lattice graphs, the cocktail party graphs, the Petersen graph, the complement of the Clebsch graph, the Shrikhande graph, the complement of the Schläfli graph, or the three Chang graphs. Switching a graph with respect to a set Y of vertices replaces all edges between Y and its complement with non-edges, and leaves edges within Y and outside Y unchanged. The set of all graphs on a vertex set X falls into equivalence classes of size , where two graphs are equivalent if one of them can be obtained from the other by switching with respect to some subset Y. The triangular graph T(m) where m ≥ 4 is the graph with vertex set the 2- element subsets of the set {1, 2, …, m} where two vertices are adjacent if and only if they are not disjoint. -
THE SHRIKHANDE GRAPH 1. Introduction a Well Studied And
THE SHRIKHANDE GRAPH RYAN M. PEDERSEN Abstract. In 1959 S.S. Shrikhande wrote a paper concerning L2 association schemes [11]. Out of this paper arose a strongly reg- ular graph with parameters (16, 6, 2, 2) that was not isomorphic to L2(4). This graph turned out to be important in the study of strongly regular graphs as a whole. In this paper, we survey the various constructions and properties of this graph. 1. Introduction A well studied and simple family of strongly regular graphs are called the square lattice graphs L2(n). These graphs have parame- ters (n2, 2(n − 1), n − 2, 2). Now strongly regular graphs with these parameters are unique for all n except n = 4. However, when n = 4 we have two non-isomorphic strongly regular graphs with parameters (16, 6, 2, 2). The non-lattice graph with these parameters is known as the Shrikhande graph. In what follows, we consider various construc- tions of this graph, along with a discussion of its various properties. 2. Constructions 2.1. The Original Construction. We begin by describing what is done in the original paper [11]. Note first that a strongly regular graph is equivalent to a two-class association scheme. Now if we can arrange the v vertices (points) into b subsets (blocks) such that Date: November 16, 2007. 1 2 RYAN M. PEDERSEN (a) Each block contains k points (all different), (b) Each point is contained in r blocks, (c) if any two points are ith associates (i = 1, 2) then they occur together in λi blocks, then we call this design D a partially balanced incomplete block (PBIB) design .