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

There appeared in 1976 an expository paper by the present author [52] entitled "What is a group ringV This question, rhetorical as it is, may nevertheless be answered directly by saying that for a group G over an integral domain R the group ring R(G) is a free unitary iî-module over the elements of G as a basis and in which the multiplication on G is extended linearly to yield an associative multiplication on R(G), R(G) becoming a ring with an identity. While this may answer the question, the underlying aim of the author is evidently to draw attention to this particular ring R(G) which, over the past decade and especially when G is infinite, has come to be intensively studied [51]. In the main R(G) is studied under the assumption that R is a field K and so, although K(G) is nowadays commonly called a group ring, K{G) in an older and more informative terminology is a linear associative algebra. The group ring K(G) of a finite group G over a field of characteristic 0 is semisimple. Over a sufficiently large extension K of the rational field Q there is a well-known theory of group characters by whose means, for example, explicit characterisations of the primitive central idempotents of K(G) are obtainable. Over a field of prime characteristic p and for G finite the Jacobson radical JK(G) of K(G) may be nontrivial but, since around 1940 [7], the development of Brauer's theory of modular characters has again, for a sufficiently large extension of the prime field GF(p), yielded characterisations of the primitive central idempotents [44]. All of this work, for which the text of Curtis and Reiner is a well-known reference [10], depends heavily on the finiteness of G. Passman's book is concerned with the case of a group ring K(G) in which G is potentially infinite and for which, in consequence, ordinary or modular character theory is of little help. The bulk of the work on infinite group rings has been done in the period 1967-1977, a major, if not the major, contributor being the present author. Prior to the mid-1960s the earliest significant work was due to Jennings [33], who for a finite /?-group G over a field K of characteristic p obtained group-theoretic descriptions of the dimension subgroups Dn(K(G)) formally defined from the ring structure as

The Algebraic Structure of Group Rings

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important special case The even-dimensional case The odd-dimensional case The bounded odd-dimensional case The bounded even-dimensional case Completion of the proof Patterns of application: Manifold structures on Poincare complexes Applications to submanifolds Submanifolds: Other techniques Separating submanifolds Two-sided submanifolds One-sided submanifolds Calculations and applications: Calculations: Surgery obstruction groups Calculations: The surgery obstructions Applications: Free actions on spheres General remarks An extension of the Atiyah-Singer $G$-signature theorem Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces Applications: Free uniform actions on euclidean space Fake tori Polycyclic groups Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work Function space methods Topological manifolds Poincare embeddings Homotopy and simple homotopy Further calculations Sullivan's results Reformulations of the algebra Rational surgery References Index.

Surgery on compact manifolds

In this paper we present several of the salient theoretical and practical issues associated with modeling a speech signal as a probabilistic function of a (hidden) Markov chain. First we give a concise review of the literature with emphasis on the Baum-Welch algorithm. This is followed by a detailed discussion of three issues not treated in the literature: alternatives to the Baum-Welch algorithm; critical facets of the implementation of the algorithms, with emphasis on their numerical properties; and behavior of Markov models on certain artificial but realistic problems. Special attention is given to a particular class of Markov models, which we call “left-to-right” models. This class of models is especially appropriate for isolated word recognition. The results of the application of these methods to an isolated word, speaker-independent speech recognition experiment are given in a companion paper.

An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

D. S. Passman

Papers

The Algebraic Structure of Group Rings

Character Theory of Finite Groups

Infinite crossed products

Units in integral group rings

Infinite group rings