Topic
Strongly regular graph
About: Strongly regular graph is a research topic. Over the lifetime, 2480 publications have been published within this topic receiving 41885 citations. The topic is also known as: strongly-regular graph.
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01 Jan 2012
TL;DR: This book gives an elementary treatment of the basic material about graph Spectra, both for ordinary, and Laplace and Seidel spectra, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics.
Abstract: This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.
2,280 citations
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23 Jun 1989TL;DR: In this paper, a connected simple graph with vertex set X of diameter d is considered, and the authors define Ri X2 by (x, y) Ri whenever x and y have graph distance.
Abstract: Consider a connected simple graph with vertex set X of diameter d. Define Ri X2 by (x, y) Ri whenever x and y have graph distance
2,264 citations
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TL;DR: In this article, it was shown that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.
919 citations
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01 Jan 1999
TL;DR: This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results.
Abstract: This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are also presented, as well as various generalisations to random graphs with given degree sequence. A major feature in this area is the small subgraph conditioning method. When applicable, this establishes a relationship between random regular graphs with uniform distribution, and non-uniform models of random regular graphs in which the probability of a graph G is weighted according to the number of subgraphs G has of a certain type. Information can be obtained in this way on the probability of existence of various types of spanning subgraphs, such as Hamilton cycles and decompositions into perfect matchings. Uniformly distributed labelled random regular graphs receive most of the attention, but also included are several non-uniform models which come about in a natural way. Some of these appear as spin-offs from the small subgraph conditioning method, and some arise from algorithms which use simple approaches to generating random regular graphs. A quite separate role played by algorithms is in the derivation of random graph properties by analysing the performance of an appropriate greedy algorithm on a random regular graph. Many open problems and conjectures are given.
692 citations
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TL;DR: In this article, the concept of a partial geometry is introduced, which serves to unify and generalize certain theorems on embedding of nets, and uniqueness of association schemes of partially balanced designs, by Bruck, Connor, Shrikhande and others.
Abstract: This paper introduced the concept of a partial geometry, which serves to unify and generalize certain theorems on embedding of nets, and uniqueness of association schemes of partially balanced designs, by Bruck, Connor, Shrikhande and others. Certain lemmas and theorems are direct generalizations of those proved by Bruck ;-57, for the case of nets, which are a special class of partial geometries.
502 citations