Topics in Algebraic Graph Theory
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Algebraic Combinatorics and Finite Geometry
An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) Algebraic Combinatorics and Finite Geometry Leo Storme Ghent University Department of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium Francqui Foundation, May 5, 2021 Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) ACKNOWLEDGEMENT Acknowledgement: A big thank you to Ferdinand Ihringer for allowing me to use drawings and latex code of his slide presentations of his lectures for Capita Selecta in Geometry (Ghent University). Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) DEFINITION A graph Γ = (X; ∼) consists of a set of vertices X and an anti-reflexive, symmetric adjacency relation ∼ ⊆ X × X. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Arxiv:1306.4805V3 [Math.OC] 6 Feb 2015 Used Greedy Techniques to Reorder Matrices
CONVEX RELAXATIONS FOR PERMUTATION PROBLEMS FAJWEL FOGEL, RODOLPHE JENATTON, FRANCIS BACH, AND ALEXANDRE D’ASPREMONT ABSTRACT. Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seri- ation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relax- ations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data. 1. INTRODUCTION We study optimization problems written over the set of permutations. While the relaxation techniques discussed in what follows are applicable to a much more general setting, most of the paper is centered on the seriation problem: we are given a similarity matrix between a set of n variables and assume that the variables can be ordered along a chain, where the similarity between variables decreases with their distance within this chain. The seriation problem seeks to reconstruct this linear ordering based on unsorted, possibly noisy, pairwise similarity information. This problem has its roots in archeology [Robinson, 1951] and also has direct applications in e.g. -
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. -
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. -
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. -
Doubly Stochastic Matrices Whose Powers Eventually Stop
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 330 (2001) 25–30 www.elsevier.com/locate/laa Doubly stochastic matrices whose powers eventually stopୋ Suk-Geun Hwang a,∗, Sung-Soo Pyo b aDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang, South Korea Received 22 June 2000; accepted 14 November 2000 Submitted by R.A. Brualdi Abstract In this note we characterize doubly stochastic matrices A whose powers A, A2,A3,... + eventually stop, i.e., Ap = Ap 1 =···for some positive integer p. The characterization en- ables us to determine the set of all such matrices. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A51 Keywords: Doubly stochastic matrix; J-potent 1. Introduction Let R denote the real field. For positive integers m, n,letRm×n denote the set of all m × n matrices with real entries. As usual let Rn denote the set Rn×1.We call the members of Rn the n-vectors. The n-vector of 1’s is denoted by e,andthe identity matrix of order n is denoted by In. For two matrices A, B of the same size, let A B denote that all the entries of A − B are nonnegative. A matrix A is called nonnegative if A O. A nonnegative square matrix is called a doubly stochastic matrix if all of its row sums and column sums equal 1. -
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. -
Arxiv:2101.02451V1 [Math.CO] 7 Jan 2021 the Diagonal Graph
The diagonal graph R. A. Bailey and Peter J. Cameron School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, U.K. Abstract According to the O’Nan–Scott Theorem, a finite primitive permu- tation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer m and group G (finite or in- finite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set Ω, whose automorphism group is the correspond- ing diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal m graph ΓD(G, m) is a Cayley graph for the group G , and so is vertex- transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the M¨obius function of the arXiv:2101.02451v1 [math.CO] 7 Jan 2021 diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory. -
Small Integral Trees
Small integral trees A. E. Brouwer Dept. of Math., Techn. Univ. Eindhoven P.O. Box 513, 5600MB Eindhoven, Netherlands [email protected] Submitted: Dec 31, 2007; Accepted: Jan 18, 2008; Published: Jan 28, 2008 Mathematics Subject Classification: 05C05, 05C30, 05E99 Abstract We give a table with all integral trees on at most 50 vertices, and characterize integral trees with a single eigenvalue 0. 1 Integral trees A finite graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues. A tree is a connected undirected graph without cycles. In this note we give a table with all integral trees on at most 50 vertices, and a further table with all known integral trees on at most 100 vertices. (For an on-line version, possibly with updates, see [1].) In particular, we find the smallest integral trees of diameter 6, and the smallest known integral tree of diameter 8. The nicest result about integral trees is that by Watanabe [12] that says that an integral tree different from K2 does not have a complete matching. Here we give a gener- alization. All `starlike' integral trees, that is, all integral trees with at most one vertex of degree larger than 2, were given by Watanabe and Schwenk [13]. All integral trees of diameter at most 3 were given in [13, 4]. See also [10, 3]. Several people have worked on constructing integral trees with large diameter, and examples with diameters 0{8 and 10 are known, see [13, 9, 3, 8, 6, 7]. It is unknown whether integral trees can have arbitrarily large diameter. -
PDF Reference
NetworkX Reference Release 1.2 Aric Hagberg, Dan Schult, Pieter Swart August 01, 2010 CONTENTS 1 Introduction 1 1.1 Who uses NetworkX?..........................................1 1.2 The Python programming language...................................1 1.3 Free software...............................................1 1.4 Goals...................................................1 1.5 History..................................................2 2 Overview 3 2.1 NetworkX Basics.............................................3 2.2 Nodes and Edges.............................................4 3 Graph types 9 3.1 Which graph class should I use?.....................................9 3.2 Basic graph types.............................................9 4 Algorithms 133 4.1 Bipartite................................................. 133 4.2 Blockmodeling.............................................. 135 4.3 Boundary................................................. 136 4.4 Centrality................................................. 137 4.5 Clique.................................................. 145 4.6 Clustering................................................ 147 4.7 Components............................................... 150 4.8 Cores................................................... 158 4.9 Cycles.................................................. 158 4.10 Directed Acyclic Graphs......................................... 159 4.11 Distance Measures............................................ 161 4.12 Eulerian................................................. -
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.