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

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

Distance-Regular 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

https://www.math.ucdavis.edu/~strohmer/papers/2002/grass.pdf

Grassmannian frames with applications to coding and communication

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

This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , any k -dimensional linear subspace R k of R r is called a ( k−1 )-dimensional elliptic subspace E k−1 . The query for equilateral point sets in elliptic geometry leads to the search for matrices B of order n and elements whose smallest eigenvalue has a high multiplicity. For n elliptic points A 1 , A 2 , …, A n , carried by the unit vectors a 1 , …, a n and spanning elliptic space E r−1 , the Gram matrix is symmetric, semipositive definite, and of rank r . B -matrices of order n ≡ 2r that have only two distinct eigenvalues with equal multiplicities r are called C -matrices. In view of the existence of a Hadamard matrix of order 92, it is interesting to know whether Paley's construction may be reversed to obtain a C -matrix of order 46.

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Equilateral point sets in elliptic geometry

MATHEMATICSEquilateral point sets in elliptic geometry

This chapter explains the perfect ternary 2-error-correcting code This code was discovered by Golay The code is connected to the theory of confounding and fractional replication In terms of coding theory, the vectors are called the words, the number of nonzero coordinates of a word its weight, the subspace generated by G* a code, the vectors of a code its codewords, G* a generator matrix, and a matrix H* generating the orthogonal complement of the code a parity check matrix of the code The fact that every 11-dimensional vector over GF(3) differs from a unique codeword makes two coordinates code a perfect code The perfect ternary Golay code partitions the vector space V(11, 3) into 3 5 cosets obtained by adding a fixed word to all codewords Each word of weight < 2 should be in a coset containing no other word of weight < 2, because the minimum nonzero weight of the code equals 5 Therefore, the 243 cosets of the code are uniquely represented by the 220 words of weight 2, the 22 words of weight 1, and the word of weight 0 The cosets form a linear space of dimension 5 over GF(3)

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J.J. Seidel

Papers

Equilateral point sets in elliptic geometry

MATHEMATICSEquilateral point sets in elliptic geometry

A strongly regular graph derived from the perfect ternary Golay code

Colloquium discrete wiskunde