A Note on the Double Total Graph T_u(Γ(R)) and T_u(Γ(Z_n×Z_m))

Abstract

Considering a commutative ring R with unity as the set of vertices and two vertices x and y are adjacent if and only if u + (x + y) ∈ Z(R) for some u ∈ U(R), the resulting graph T u (Γ(R)) is known as the double total graph. In this paper we find the degree of any vertex in T u (Γ(R)) for a weakly unit fusible ring R and domination number of T u (Γ(R)) for any ring R. Also, we investigate the properties of T u (Γ(Z n × Z m )) and characterize R in terms of toroidal T u (Γ(R)).