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 |
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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 |

Related URLs: | |

Depositing User: | LivePure Connector |

Date Deposited: | 11 Dec 2018 14:30 |

Last Modified: | 22 Oct 2022 04:17 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/69278 |

DOI: | 10.1007/s10801-018-0860-0 |

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