Abstract :
Let G be a finite group. An element g∈G is called non-vanishing, if for every irreducible complex character χ of G, χ(g)≠0. The bi-Cayley graph BCay(G,T) of G with respect to a subset T⊆G, is an undirected graph with vertex set G×{1,2} and edge set {{(x,1),(tx,2)}∣x∈G, t∈T}. Let nv(G) be the set of all non-vanishing elements of a finite group G. We show that g∈nv(G) if and only if the adjacency matrix of BCay(G,T), where T=Cl(g) is the conjugacy class of g, is non-singular. We prove that if the commutator subgroup of G has prime order p, then (1) g∈nv(G) if and only if |Cl(g)| p, (2) if p is the smallest prime divisor of |G|, then nv(G)=Z(G). Also we show that (a) if Cl(g)={g,h}, then g∈nv(G) if and only if gh^−1 has odd order, (b) if |Cl(g)|∈{2,3} and (ord(g),6)=1, then g∈nv(G).
Keywords :
Non , vanishing element , character , conjugacy class , Bi , Cayley graph
