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A classification of all 1-Salem graphs

Part of: Algebraic number theory: global fields Graph theory

Published online by Cambridge University Press:  01 November 2014

Lee Gumbrell  and
James McKee
Lee Gumbrell
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, United Kingdom email Lee.Gumbrell@gmail.com
James McKee
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, United Kingdom email James.McKee@rhul.ac.uk

Abstract

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One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$. On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.

We define an $m$-Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$. The $1$-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$-Salem graph as an induced subgraph, so these $1$-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$-Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 582 - 594
Copyright
© The Author(s) 2014 

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