Remarks on singular Cayley graphs and vanishing elements of simple groups

Siemons, Johannes and Zalesski, Alexandre (2019) Remarks on singular Cayley graphs and vanishing elements of simple groups. Journal of Algebraic Combinatorics, 50 (4). pp. 379-401. ISSN 0925-9899

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Abstract

Let Γ be a finite graph and let A(Γ) be its adjacency matrix. Then Γ is singular if A(Γ) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H) when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ of G for which ∑h∈Hχ(h)=0. At this stage, we focus on the case when H is a single conjugacy class hG of G; in this case, the above equality is equivalent to χ(h)=0 . Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G is called vanishing if χ(h)=0 for some irreducible character χ of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.

Item Type: Article
Uncontrolled Keywords: singular cayley graphs,vertex transitive graphs,vanishing elements,block theory of symmetric and alternating groups
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: LivePure Connector
Date Deposited: 11 Dec 2018 14:30
Last Modified: 18 Mar 2020 02:14
URI: https://ueaeprints.uea.ac.uk/id/eprint/69278
DOI: 10.1007/s10801-018-0860-0

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