Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.

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Algebraic Graph Theory

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.

Spectra of graphs

This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference.

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A course in combinatorics

Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension d ( X ) of the space spanned by X , its cardinality n = | X |, its degree s( X ), and its strength t ( X ).The chapter presents derivation of bounds for the cardinality of spherical A -codes in terms of the Gegenbauer coefficients of polynomials compatible with A . It also discusses spherical ( d , n , s , i)- configurations X . These are sets X of cardinality n on the unit sphere Ω d , which are spherical t -designs and spherical A -codes with I A I = s ; in other words, the strength t ( X ) is at least t and the degree s ( X ) is at most s . A condition is given for a spherical A -code to be a spherical t -design, in terms of the Gegenbauer coefficients of an annihilator of the set A . The chapter presents many examples of spherical ( d , n , s , t )-configurations; there exist tight spherical t-designs with t = 2, 3, 4, 5, 7, 11, and non-tight spherical ( 2s − 1)-designs. The constructions of these examples use sets of lines with few angles and association schemes, respectively.

Spherical codes and designs

Bilinear forms over a finite field, with applications to coding theory

in whole or in part is permitted for any purpose of the United states Government.

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On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs

t-designs on hypergraphs

Intersections among Steiner systems

The Connectivity of Strongly Regular Graphs

An extensive family of two-class PBIB designs is defined in Section 1 and given the title of NL designs. Several analogies of NL designs with Latin square type designs are discussed in Sections 1 and 2. Some of these designs have already appeared in scattered contexts. Section 3 gives a construction based on finite geometry for an infinite class of NL designs, and others can be constructed by methods to be taken up elsewhere. An extension to designs with more than two associate classes is indicated in Section 4.

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Dale M. Mesner

Papers

On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs

t-designs on hypergraphs

Intersections among Steiner systems

The Connectivity of Strongly Regular Graphs

A New Family of Partially Balanced Incomplete Block Designs with Some Latin Square Design Properties