Locally $ S $-Distance Transitive Graphs
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Interval Edge-Colorings of Graphs
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2016 Interval Edge-Colorings of Graphs Austin Foster University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Foster, Austin, "Interval Edge-Colorings of Graphs" (2016). Electronic Theses and Dissertations, 2004-2019. 5133. https://stars.library.ucf.edu/etd/5133 INTERVAL EDGE-COLORINGS OF GRAPHS by AUSTIN JAMES FOSTER B.S. University of Central Florida, 2015 A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Summer Term 2016 Major Professor: Zixia Song ABSTRACT A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher con- ferences so that every person’s conferences occur in consecutive slots. -
On the Metric Dimension of Imprimitive Distance-Regular Graphs
On the metric dimension of imprimitive distance-regular graphs Robert F. Bailey Abstract. A resolving set for a graph Γ is a collection of vertices S, cho- sen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric di- mension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance- regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible. Mathematics Subject Classification (2010). Primary 05E30; Secondary 05C12. Keywords. Metric dimension; resolving set; distance-regular graph; imprimitive; halved graph; folded graph; bipartite double; Taylor graph; incidence graph. 1. Introduction Let Γ = (V; E) be a finite, undirected graph without loops or multiple edges. For u; v 2 V , the distance from u to v is the least number of edges in a path from u to v, and is denoted dΓ(u; v) (or simply d(u; v) if Γ is clear from the context). A resolving set for a graph Γ = (V; E) is a set of vertices R = fv1; : : : ; vkg such that for each vertex w 2 V , the list of distances (d(w; v1);:::; d(w; vk)) uniquely determines w. -
Uniform Factorizations of Graphs
Western Michigan University ScholarWorks at WMU Dissertations Graduate College 12-1979 Uniform Factorizations of Graphs David Burns Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Mathematics Commons Recommended Citation Burns, David, "Uniform Factorizations of Graphs" (1979). Dissertations. 2682. https://scholarworks.wmich.edu/dissertations/2682 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. UNIFORM FACTORIZATIONS OF GRAPHS by David Burns A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the Degree of Doctor of Philosophy Western Michigan University Kalamazoo, Michigan December 197 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT S I wish to thank Professor Shashichand Kapoor, my advisor, for his guidance and encouragement during the writing of this dissertation. I would also like to thank my other working partners Professor Phillip A. Ostrand of the University of California at Santa Barbara and Professor Seymour Schuster of Carleton College for their many contributions to this study. I am also grateful to Professors Yousef Alavi, Gary Chartrand, and Dionsysios Kountanis of Western Michigan University for their careful reading of the manuscript and to Margaret Johnson for her expert typing. David Burns Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. -
A Characterization of the Doubled Grassmann Graphs, the Doubled Odd Graphs, and the Odd Graphs by Strongly Closed Subgraphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 24 (2003) 161–171 www.elsevier.com/locate/ejc A characterization of the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs by strongly closed subgraphs Akira Hiraki Division of Mathematical Sciences, Osaka Kyoiku University, 4-698-1 Asahigaoka, Kashiwara, Osaka 582-8582, Japan Received 27 November 2001; received in revised form 14 October 2002; accepted 3 December 2002 Abstract We characterize the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs by the existence of sequences of strongly closed subgraphs. c 2003 Elsevier Science Ltd. All rights reserved. 1. Introduction Known distance-regular graphs often have many subgraphs with high regularity. For example the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs satisfy the following condition. For any given pair (u,v) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance of u and v. In particular, any ‘non- trivial’ strongly closed subgraph is distance-biregular, and that is regular iff its diameter is odd. Our purpose in this paper is to prove that a distance-regular graph with this condition is either the doubled Grassmann graph, the doubled Odd graph, or the Odd graph. First we recall our notation and terminology. All graphs considered are undirected finite graphs without loops or multiple edges. Let Γ be a connected graph with usual distance ∂Γ . We identify Γ with the set of vertices. -
Tilburg University Graphs Whose Normalized Laplacian Has Three
Tilburg University Graphs whose normalized laplacian has three eigenvalues van Dam, E.R.; Omidi, G.R. Published in: Linear Algebra and its Applications Publication date: 2011 Document Version Peer reviewed version Link to publication in Tilburg University Research Portal Citation for published version (APA): van Dam, E. R., & Omidi, G. R. (2011). Graphs whose normalized laplacian has three eigenvalues. Linear Algebra and its Applications, 435(10), 2560-2569. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 29. sep. 2021 Graphs whose normalized Laplacian has three eigenvalues¤ E.R. van Dama G.R. Omidib;c;1 aTilburg University, Dept. Econometrics and Operations Research, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands bDept. Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran cSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran edwin:vandam@uvt:nl romidi@cc:iut:ac:ir Abstract We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. -
A Godsil E Thomason I Solomon M Abiad B Van Dam F Balbuena J
8.30–9.15 am A Godsil E Thomason I Solomon M Abiad 9.20–9.40 am 1 Zhan 10 Bukh 19 Srinivasan 24 Reichard 9.45–10.05 am 2 Ye 11 Martin 20 Sumalroj 25 Xu 10.05–10.35 am break break break break 10.35–10.55 am 3 Dalfó 12 Kamat 21 Bencs 26 Peng 11.00–11.20 am 4 McGinnis 13 Timmons 22 Guo 27 Kravitz 11.25–12.10 pm B Van Dam F Balbuena J Muzychuk N Xiang 12.10–2.00 pm lunch lunch lunch lunch 2.00–2.45 pm C Kantor G Füredi K Williford 2.50–3.10 pm 5 Gu 14 Tait 23 Ducey 3.15–3.35 pm 6 Coutinho 15 Kodess L Woldar break 3.40–4.00 pm 7 Greaves 16 Y. Wang 4.00–4.30 pm break break 4.30–4.50 pm 8 Fiol 17 Kronenthal 4:55–5.15 pm 9 W. Wang 18 Moorhouse 5.20–6.10 pm D Haemers H Lazebnik 2 8.30–9.15 am A Chris Godsil Spectral invariants from embeddings 9.20–9.40 am 1 Harmony Zhan Quantum walks and mixing 9.45–10.05 am 2 Dong Ye Median eigenvalues and graph inverse 10.35–10.55 am 3 Cristina Dalfó Characterizing identifying codes from the spectrum of a graph or digraph 11.00–11.20 am 4 Matt McGinnis The smallest eigenvalues of the Hamming graphs 11.25–12:10 pm B Edwin van Dam Partially metric association schemes with a small multiplicity 2.00–2.45 pm C Bill Kantor MUBs 2.50–3.10 pm 5 Xiaofeng Gu Toughness, connectivity and the spectrum of regular graphs 3.15–3.35 pm 6 Gabriel Coutinho Average mixing matrix 3.40–4.00 pm 7 Gary Greaves Edge-regular graphs and regular cliques 4.30–4.50 pm 8 Miguel Angel Fiol An algebraic approach to lifts of digraphs 4.55–5.15 pm 9 Wei Wang A positive proportion of multigraphs are determined by their generalized spectra 5.20–6.10 -
Edge Fault-Tolerance Analysis of Maximally Edge-Connected Graphs
Journal of Computer and System Sciences 115 (2021) 64–72 Contents lists available at ScienceDirect Journal of Computer and System Sciences www.elsevier.com/locate/jcss Edge fault-tolerance analysis of maximally edge-connected ✩ graphs and super edge-connected graphs ∗ Shuang Zhao a, Zongqing Chen b, Weihua Yang c, Jixiang Meng a, a College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China b School of Science, Tianjin Chengjian University, Tianjin 300384, China c Department of Mathematics, Taiyuan University of Techology, Taiyuan 030024, China a r t i c l e i n f o a b s t r a c t Article history: Edge fault-tolerance of interconnection network is of significant important to the design Received 8 October 2019 and maintenance of multiprocessor systems. A connected graph G is maximally edge- Received in revised form 20 May 2020 connected (maximally-λ for short) if its edge-connectivity attains its minimum degree. G is Accepted 8 July 2020 super edge-connected (super-λ for short) if every minimum edge-cut isolates one vertex. Available online 23 July 2020 The edge fault-tolerance of the maximally-λ (resp. super-λ) graph G with respect to the Keywords: maximally-λ (resp. super-λ) property, denoted by mλ(G) (resp. Sλ(G)), is the maximum − Edge fault-tolerance integer m for which G S is still maximally-λ (resp. super-λ) for any edge subset S Maximally edge-connected with |S| ≤ m. In this paper, we give upper and lower bounds on mλ(G). Furthermore, we Super edge-connected completely determine the exact values of mλ(G) and Sλ(G) for vertex transitive graphs. -
Distance-Transitive Graphs
Distance-Transitive Graphs Submitted for the module MATH4081 Robert F. Bailey (4MH) Supervisor: Prof. H.D. Macpherson May 10, 2002 2 Robert Bailey Department of Pure Mathematics University of Leeds Leeds, LS2 9JT May 10, 2002 The cover illustration is a diagram of the Biggs-Smith graph, a distance-transitive graph described in section 11.2. Foreword A graph is distance-transitive if, for any two arbitrarily-chosen pairs of vertices at the same distance, there is some automorphism of the graph taking the first pair onto the second. This project studies some of the properties of these graphs, beginning with some relatively simple combinatorial properties (chapter 2), and moving on to dis- cuss more advanced ones, such as the adjacency algebra (chapter 7), and Smith’s Theorem on primitive and imprimitive graphs (chapter 8). We describe four infinite families of distance-transitive graphs, these being the Johnson graphs, odd graphs (chapter 3), Hamming graphs (chapter 5) and Grass- mann graphs (chapter 6). Some group theory used in describing the last two of these families is developed in chapter 4. There is a chapter (chapter 9) on methods for constructing a new graph from an existing one; this concentrates mainly on line graphs and their properties. Finally (chapter 10), we demonstrate some of the ideas used in proving that for a given integer k > 2, there are only finitely many distance-transitive graphs of valency k, concentrating in particular on the cases k = 3 and k = 4. We also (chapter 11) present complete classifications of all distance-transitive graphs with these specific valencies. -
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 . -
On Distance, Geodesic and Arc Transitivity of Graphs
On distance, geodesic and arc transitivity of graphs Alice Devillers, Wei Jin,∗ Cai Heng Li and Cheryl E. Praeger†‡ Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia Abstract We compare three transitivity properties of finite graphs, namely, for a positive integer s, s-distance transitivity, s-geodesic transitivity and s-arc transitivity. It is known that if a finite graph is s-arc transi- tive but not (s + 1)-arc transitive then s ≤ 7 and s 6= 6. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of s, and that these graphs can have arbitrar- ily large diameter if and only if 1 ≤ s ≤ 3. Moreover, for a prime p we prove that there exists a graph of valency p that is 2-geodesic transitive but not 2-arc transitive if and only if p ≡ 1 (mod 4), and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph Kp+1 and is geodesic transitive with automorphism group PSL(2,p) × Z2. Keywords: Graphs; distance-transitivity; geodesic-transitivity; arc-transitivity 1 Introduction arXiv:1110.2235v1 [math.CO] 11 Oct 2011 The study of finite distance-transitive graphs goes back to Higman’s paper [9] in which “groups of maximal diameter” were introduced. These are per- mutation groups which act distance transitively on some graph. The family ∗The second author is supported by the Scholarships for International Research Fees (SIRF) at UWA. -
Distance-Regular Cayley Graphs with Small Valency∗
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 203–222 https://doi.org/10.26493/1855-3974.1964.297 (Also available at http://amc-journal.eu) Distance-regular Cayley graphs with small valency∗ Edwin R. van Dam Department of Econometrics and O.R., Tilburg University, The Netherlands Mojtaba Jazaeri y Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran Received 28 March 2019, accepted 9 May 2019, published online 20 September 2019 Abstract We consider the problem of which distance-regular graphs with small valency are Cay- ley graphs. We determine the distance-regular Cayley graphs with valency at most 4, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 5, and the Cayley graphs among all distance-regular graphs with girth 3 and valency 6 or 7. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some “excep- tional” distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs. Keywords: Cayley graph, distance-regular graph. Math. Subj. Class.: 05E30 ∗We thank Sasha Gavrilyuk, who, in the final stages of writing this paper, at a conference in Plzen,ˇ pointed us to [27] for what he called Higman’s method applied to generalized polygons. -
Dissertation Maximal Curves, Zeta Functions, and Digital
DISSERTATION MAXIMAL CURVES, ZETA FUNCTIONS, AND DIGITAL SIGNATURES Submitted by Beth Malmskog Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2011 Doctoral Committe: Advisor: Rachel Pries Jeffrey Achter Tim Penttila Jacob Roberts ABSTRACT MAXIMAL CURVES, ZETA FUNCTIONS, AND DIGITAL SIGNATURES Curves with as many points as possible over a finite field Fq under the Hasse- Weil bound are called maximal curves. Besides being interesting as extremal objects, maximal curves have applications in coding theory. A maximal curves may also have a great deal of symmetry, i.e. have an automorphism group which is large compared to the curve's genus. In Part 1, we study certain families of maximal curves and find a large subgroup of each curve's automorphism group. We also give an upper bound for the size of the automorphism group. In Part 2, we study the zeta functions of graphs. The Ihara zeta function of a graph was defined by Ihara in the 1960s. It was modeled on other zeta functions in its form, an infinite product over primes, and has some analogous properties, for example convergence to a rational function. The knowledge of the zeta function of a regular graph is equivalent to knowledge of the eigenvalues of its adjacency matrix. We calculate the Ihara zeta function for an infinite family of irregular graphs and consider how the same technique could be applied to other irregular families. We also discuss ramified coverings of graphs and a joint result with Michelle Manes on the divisibility properties of zeta functions for graphs in ramified covers.