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.
Original language  English 

Pages (fromto)  379401 
Number of pages  23 
Journal  Journal of Algebraic Combinatorics 
Volume  50 
Issue number  4 
Early online date  30 Nov 2018 
DOIs  
Publication status  Published  Dec 2019 
Keywords
 Singular Cayley graphs
 Vertex transitive graphs
 Vanishing elements
 Block theory of symmetric and alternating groups
Profiles

Johannes Siemons
 School of Mathematics  Emeritus Reader
 Algebra and Combinatorics  Member
Person: Honorary, Research Group Member