Which elements of a finite group are non-vanishing?

‎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)‎.