Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity
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Results on Domination Number and Bondage Number for Some Families of Graphs
International Journal of Pure and Applied Mathematics Volume 107 No. 3 2016, 565-577 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v107i3.5 ijpam.eu RESULTS ON DOMINATION NUMBER AND BONDAGE NUMBER FOR SOME FAMILIES OF GRAPHS G. Hemalatha1, P. Jeyanthi2 § 1Department of Mathematics Shri Andal Alagar College of Engineering Mamandur, Kancheepuram, Tamil Nadu, INDIA 2Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur 628 215, Tamil Nadu, INDIA Abstract: Let G = (V, E) be a simple graph on the vertex set V . In a graph G, A set S ⊆ V is a dominating set of G if every vertex in VS is adjacent to some vertex in S. The domination number of a graph Gγ(G)] is the minimum size of a dominating set of vertices in G. The bondage number of a graph G[Bdγ(G)] is the cardinality of a smallest set of edges whose removal results in a graph with domination number larger than that of G. In this paper we establish domination number and the bondage number for some families of graphs. AMS Subject Classification: 05C69 Key Words: dominating set, bondage number, domination number, cocktail party graph, coxeter graph, crown graph, cubic symmetric graph, doyle graph, folkman graph, levi graph, icosahedral graph 1. Introduction The domination in graphs is one of the major areas in graph theory which attracts many researchers because it has the potential to solve many real life problems involving design and analysis of communication network, social net- Received: October 17, 2015 c 2016 Academic Publications, Ltd. -
TETRAVALENT SYMMETRIC GRAPHS of ORDER 9P
J. Korean Math. Soc. 49 (2012), No. 6, pp. 1111–1121 http://dx.doi.org/10.4134/JKMS.2012.49.6.1111 TETRAVALENT SYMMETRIC GRAPHS OF ORDER 9p Song-Tao Guo and Yan-Quan Feng Abstract. A graph is symmetric if its automorphism group acts transi- tively on the set of arcs of the graph. In this paper, we classify tetravalent symmetric graphs of order 9p for each prime p. 1. Introduction Let G be a permutation group on a set Ω and α ∈ Ω. Denote by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α. We say that G is semiregular on Ω if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular. Throughout this paper, we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X), E(X) and Aut(X) to denote its vertex set, edge set, and automorphism group, respectively. For u , v ∈ V (X), denote by {u , v} the edge incident to u and v in X. A graph X is said to be vertex-transitive if Aut(X) acts transitively on V (X). An s-arc in a graph is an ordered (s + 1)-tuple (v0, v1,...,vs−1, vs) of vertices of the graph X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s, and vi−1 = vi+1 for 1 ≤ i ≤ s − 1. In particular, a 1-arc is called an arc for short and a 0-arc is a vertex. -
Constructing Arbitrarily Large Graphs with a Specified Number Of
Electronic Journal of Graph Theory and Applications 4 (1) (2019), 1–8 Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles Michael Haythorpea aSchool of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia michael.haythorpe@flinders.edu.au Abstract A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing 5n − 9 vertices and k Hamiltonian cycles for any choice of integers n ≥ k ≥ 4. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen to be much smaller than n. Keywords: Hamiltonian cycles, Graph Construction, Broken Crown Mathematics Subject Classification : 05C45 1. Introduction The Hamiltonian cycle problem (HCP) is a famous NP-complete problem in which one must determine whether a given graph contains a simple cycle traversing all vertices of the graph, or not. Such a simple cycle is called a Hamiltonian cycle (HC), and a graph containing at least one arXiv:1902.10351v1 [math.CO] 27 Feb 2019 Hamiltonian cycle is said to be a Hamiltonian graph. Typically, randomly generated graphs (such as Erdos-R˝ enyi´ graphs), if connected, are Hamil- tonian and contain many Hamiltonian cycles. Although HCP is an NP-complete problem, for these graphs it is often fairly easy for a sophisticated heuristic (e.g. see Concorde [1], Keld Helsgaun’s LKH [4] or Snakes-and-ladders Heuristic [2]) to discover one of the multitude of Hamiltonian cy- cles through a clever search. -
Prime Labeling of Union Graphs
International Journal of Scientific Research in _____________________________ Research Paper . Mathematical and Statistical Sciences Vol.6, Issue.1, pp.86-93, February (2019) E-ISSN: 2348-4519 DOI: https://doi.org/10.26438/ijsrmss/v6i1.8693 Prime labeling of Union Graphs S K Patel1, A N Kansagara2* 1 Department of Mathematics, Government Engineering College, Bhuj-370 001. India. 2 Department of Mathematics, Shree D K V Arts and Science College, Jamnagar-361 008. India. *Corresponding author: [email protected] Tel: +91 94267 57504 Available online at: www.isroset.org Received: 01/Feb/2019, Accepted: 11/Feb/2019, Online: 28/Feb/2019 Abstract- A graph with vertices is called prime graph if its vertices can be labeled with first positive integers such that each pair of adjacent vertices have relatively prime labels. Such a labeling is called prime labeling. It is known that graphs like gear graphs, crown graphs, helm graphs, book graphs and are prime. In this paper, we study the prime labeling of union of these graphs. Keyword: Prime labeling, union of graphs, Corona product, Cartesian Product. AMS Subject Classification(2010): 05C7 I. INTRODUCTION (iii) whereas and are odd integers (iv) , where is any integer. All graphs considered in this paper are simple, finite, and 2. undirected. For a graph , and denotes the 3. , where is an odd integer. vertex set and edge set of respectively whereas 4. For arbitrary integers if and denotes the cardinality of the respective sets. For , then . terms not defined here, we refer the reader to [6]. Paths and cycles are the immediate examples of Now we discuss some basic definions which will be used in prime graphs but on the other hand the conjecture that trees main results. -
Smith Forms for Adjacency Matrices of Circulant Graphs
Smith forms for adjacency matrices of circulant graphs Gerald Williamsa aDepartment of Mathematical Sciences, University of Essex, Colchester, Essex CO4 3SQ, U.K. Abstract We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andr´asfaigraph, Doob graph, cocktail party graph, crown graph, prism graph, M¨obiusladder. The proofs operate by finding the abelianisation of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. Keywords: Smith normal form, circulant graph, adjacency matrix. 2000 MSC: 05C50, 15A21, 20F05. 1. Introduction The circulant matrix circn(a0; : : : ; an−1) is the n × n matrix whose first row is (a0; : : : ; an−1) and where row (i+1) (0 ≤ i ≤ n−2) is a cyclic shift of row i by one column. A circulant graph is a graph that is isomorphic to a graph whose adjacency matrix is circulant. We shall write A(Γ) for the adjacency matrix of a graph Γ. Given graphs Γ; Γ0 if det(A(Γ)) =6 det(A(Γ0)) (in particular if precisely one of A(Γ), A(Γ0) is singular) then Γ; Γ0 are non-isomorphic. Similarly, if rank(A(Γ)) =6 rank(A(Γ0)) then Γ; Γ0 are non-isomorphic. Singularity, rank, and determinants of various families of circulant graphs are considered (for example) in [3],[7],[16],[23]. For an n × n integer matrix M, the Smith normal form of M, written SNF(M), is the n × n diagonal integer matrix S = diagn(d0; : : : ; dn−1) where d0; : : : ; dn−1 2 N [ f0g and dijdi+1 (0 ≤ i ≤ n − 2) is such that there exist invertible integer matrices P; Q such that PMQ = S. -
Group Actions on Graphs, Maps and Surfaces
Ministry of Education of Slovak Republic Scienti¯c Board of P. J. Saf¶arikUniversity· Doc. RNDr. Roman Nedela, CSc. Group Actions on Graphs, Maps and Surfaces Dissertation for the Degree of Doctor of Physical-Mathematical Sciences Branch: 11-11-9 Discrete Mathematics Ko·sice,May 2005 Contents 1 Introduction 5 2 Half-arc-transitive actions of groups on graphs of valency four 7 2.1 Graphs and groups of automorphisms . 7 2.2 Half-arc-transitive action, G-orientation. 7 2.3 Orbital graphs . 9 2.4 A construction of half-arc-transitive graphs of valency 4 . 10 2.5 Alternating cycles, classi¯cation of tightly attached graphs . 10 2.6 Regular maps and half-arc-transitive graphs of valency four . 11 2.7 P l and Al operators on graphs of valency 4 . 13 2.8 Graphs of valency 4 and girth 4 . 14 2.9 Classi¯cation of point stabilizers . 15 2.10 Relations to group actions of other sorts . 17 3 Maps, Regular Maps and Hypermaps 21 3.1 Topological and combinatorial maps, permutation representation of maps . 21 3.2 Generalization to hypermaps, Walsh map of a hypermap . 28 3.3 Maps, hypermaps and groups . 30 3.4 Regular maps of large planar width and residual ¯niteness of triangle groups . 37 3.5 Maps, hypermaps and Riemann surfaces . 40 3.6 Enumeration of maps of given genus . 43 3.7 Regular hypermaps on a ¯xed surface . 48 3.8 Operations on maps and hypermaps, external symmetries of hy- permaps . 51 3.9 Lifting automorphisms of maps . 54 3.10 Regular embeddings of graphs . -
Cores of Vertex-Transitive Graphs
Cores of Vertex-Transitive Graphs Ricky Rotheram Submitted in total fulfilment of the requirements of the degree of Master of Philosophy October 2013 Department of Mathematics and Statistics The University of Melbourne Parkville, VIC 3010, Australia Produced on archival quality paper Abstract The core of a graph Γ is the smallest graph Γ∗ for which there exist graph homomor- phisms Γ ! Γ∗ and Γ∗ ! Γ. Thus cores are fundamental to our understanding of general graph homomorphisms. It is known that for a vertex-transitive graph Γ, Γ∗ is vertex-transitive, and that jV (Γ∗)j divides jV (Γ)j. The purpose of this thesis is to determine the cores of various families of vertex-transitive and symmetric graphs. We focus primarily on finding the cores of imprimitive symmetric graphs of order pq, where p < q are primes. We choose to investigate these graphs because their cores must be symmetric graphs with jV (Γ∗)j = p or q. These graphs have been completely classified, and are split into three broad families, namely the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. We use this classification to determine the cores of all imprimitive symmetric graphs of order pq, using differ- ent approaches for the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. Circulant graphs are examples of Cayley graphs of abelian groups. Thus, we generalise the approach used to determine the cores of the symmetric circulants of order pq, and apply it to other Cayley graphs of abelian groups. Doing this, we show that if Γ is a Cayley graph of an abelian group, then Aut(Γ∗) contains a transitive subgroup generated by semiregular automorphisms, and either Γ∗ is an odd cycle or girth(Γ∗) ≤ 4. -
Pattern Avoidance and Fiber Bundle Structures on Schubert Varieties
Thursday 2.10 Auditorium B Pattern Avoidance and Fiber Bundle Structures on Schubert Varieties Timothy Alland [email protected] SUNY Stony Brook (This talk is based on joint work with Edward Richmond.) MSC2000: 05E15, 14M15 Permutations are known to index type A Schubert varieties. A question one can ask regarding a specific Schubert variety is whether it is an iterated fiber bundle of Grass- mannian Schubert varieties, that is, if it has a complete parabolic bundle structure. We have found that a Schubert variety has a complete parabolic bundle structure if and only if its associated permutation avoids the patterns 3412, 52341, and 635241. We find this by first identifying when the standard projection from the Schubert variety in the complete flag variety to the Schubert variety in the Grassmannian is a fiber bundle using what we have called \split pattern avoidance". In this talk, I will demonstrate how we were able to move from a characterization of this projection in terms of the support and left descents of the permutation's parabolic decomposition to one that applies split pattern avoidance. I will also give a flavor of the proof of how the three patterns mentioned above determine whether a Schubert variety has a complete parabolic bundle structure. Monday 3.55 Auditorium B A simple proof of Shamir's conjecture Peter Allen [email protected] LSE (This talk is based on joint work with Julia B¨ottcher, Ewan Davies, Matthew Jenssen, Yoshiharu Kohayakawa, Barnaby Roberts.) MSC2000: 05C80 It is well known (and easy to show) that the threshold for a perfect matching in the bino- log n mial random graph G(n; p) is p = Θ n , coinciding with the threshold for every vertex to be in an edge (and much more is known). -
Arxiv:2108.05223V2 [Math.CO] 12 Aug 2021 the Line Graph of the Crown
The line graph of the crown graph is distance integral S. Morteza Mirafzal Department of Mathematics Lorestan University, Khorramabad, Iran E-mail: [email protected] E-mail: [email protected] Abstract The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix D(G). A graph is called distance integral if all of its distance eigenvalues are integers. Let n ≥ 3 be an integer. A crown graph Cr(n) is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. Let L(Cr(n)) denote the line graph of the crown graph Cr(n). In this paper, by using the orbit partition method in algebraic graph theory, we determine the set of all distance eigenvalues of L(Cr(n)) and show that this graph is distance integral. 1 Introduction and Preliminaries arXiv:2108.05223v4 [math.CO] 8 Sep 2021 In this paper, a graph G = (V, E) is considered as an undirected simple graph where V = V (G) is the vertex-set and E = E(G) is the edge-set. For all the terminology and notation not defined here, we follow [3,4,5,6,7]. Let G = (V, E) be a graph and A = A(G) be an adjacency matrix of G. The characteristic polynomial of G is defined as P (G; x)= P (x)= |xI − A|. A zero of p(x) is called an eigenvalue of the graph G. A graph is called integral, if all the eigenvalues are integers. The study of integral graphs was initiated by 2010 Mathematics Subject Classification:05C50 Keywords: crown graph, distance integral, vertex-transitive, orbit partition Date: 1 2 S. -
Trivalent Orbit Polynomial Graphs* Robert A. Beezer Department Of
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Trivalent Orbit Polynomial Graphs* Robert A. Beezer Department of Mathematics & Computer Science University of Puget Sound Tacoma, Washington 98416 Submittedby Richard A. Brualdi ABSTRACT We introduce orbit polynomial graphs, and discuss their connections with dis- tance-transitive, distance-regular, and distance polynomial graphs. After some general results, we classify all of the nonsymmetric trivalent orbit polynomial graphs. 1. INTRODUCTION Given a finite, undirected graph I with no loops or multiple edges, there are many connections between its automorphism group, adjacency algebra, eigenvalues, and distance structure. We will introduce the notion of an orbit polynomial graph, and discuss its relationship to distance-transitive, distance- regular, and distance polynomial graphs. Then we will find all of the nonsymmetric trivalent orbit polynomial graphs. For a graph I, the adjacency matrix A( I) = (a i j) is given by 1 if {vi,vj}EEI, aij = 0 otherwise. The adjacency algebra of I, st( I), is the algebra of all polynomials in A( I). The eigenvalues of I are the eigenvalues of A(I), and the number of distinct eigenvalues of I will be denoted here as e(I). Because A(T) is a real symmetric matrix, e(I) equals the dimension of Q(r). *This work was supportedin part by the Research Board of the University of Illinois at Urbana-Champaign. LINEAR ALGEBRA AND ITS APPLZCATZONS 73:133-146 (1986) 133 0 Elsevier Science Publishing Co., Inc., 1986 52 Vanderbilt Ave., New York, NY 10017 00243795/86/$3.50 134 ROBERT A. -
An Introduction to Algebraic Graph Theory
An Introduction to Algebraic Graph Theory Cesar O. Aguilar Department of Mathematics State University of New York at Geneseo Last Update: March 25, 2021 Contents 1 Graphs 1 1.1 What is a graph? ......................... 1 1.1.1 Exercises .......................... 3 1.2 The rudiments of graph theory .................. 4 1.2.1 Exercises .......................... 10 1.3 Permutations ........................... 13 1.3.1 Exercises .......................... 19 1.4 Graph isomorphisms ....................... 21 1.4.1 Exercises .......................... 30 1.5 Special graphs and graph operations .............. 32 1.5.1 Exercises .......................... 37 1.6 Trees ................................ 41 1.6.1 Exercises .......................... 45 2 The Adjacency Matrix 47 2.1 The Adjacency Matrix ...................... 48 2.1.1 Exercises .......................... 53 2.2 The coefficients and roots of a polynomial ........... 55 2.2.1 Exercises .......................... 62 2.3 The characteristic polynomial and spectrum of a graph .... 63 2.3.1 Exercises .......................... 70 2.4 Cospectral graphs ......................... 73 2.4.1 Exercises .......................... 84 3 2.5 Bipartite Graphs ......................... 84 3 Graph Colorings 89 3.1 The basics ............................. 89 3.2 Bounds on the chromatic number ................ 91 3.3 The Chromatic Polynomial .................... 98 3.3.1 Exercises ..........................108 4 Laplacian Matrices 111 4.1 The Laplacian and Signless Laplacian Matrices .........111 4.1.1 -
Recent Trends and Future Directions in Vertex-Transitive Graphs
Also available at http://amc.imfm.si ISSN 1855-3966 (printed ed.), ISSN 1855-3974 (electronic ed.) ARS MATHEMATICA CONTEMPORANEA 1 (2008) 112–125 Recent Trends and Future Directions in Vertex-Transitive Graphs Klavdija Kutnar ∗ University of Primorska, FAMNIT, Glagoljaskaˇ 8, 6000 Koper, Slovenia Dragan Marusiˇ cˇ ∗ University of Primorska, FAMNIT, Glagoljaskaˇ 8, 6000 Koper, Slovenia University of Ljubljana, PEF, Kardeljeva plosˇcadˇ 16, 1000 Ljubljana, Slovenia Received 3 August 2008, accepted 19 August 2008, published online 22 August 2008 Abstract A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular au- tomorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade. Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group. Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs – the Hamilton path/cycle problem and the semiregularity problem – as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs.