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

On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

/pdf/the-geometry-of-two-weight-codes-nnndeshbae.pdf

The Geometry of Two-Weight Codes

LetG be a finite group of order ν. Ak-element subsetD ofG is called a (ν,k, λ, μ)-partial difference set if the expressionsgh
−1, forg andh inD withg≠h, represent each nonidentity element inD exactly λ times and each nonidentity element not inD exactly μ times. Ife∉D andg∈D iffg
−1∈D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.

A survey of partial difference sets

Caps and codes

The regular two-graph on 276 vertices

Weights of linear codes and strongly regular normed spaces

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Weights of linear codes and strongly regular normed spaces