Zeta Functions of the Complement and xyz-Transformations of a Regular Graph

Let Z(λ,G)denote the zeta function of a graph G.In this paper the complement G~Cand the G-transformation Gof an r-regular graph G with n vertices and m edges for x,y,z∈{0,1,+,-},are considerd.The relationship between Z(λ,G)and Z(λ,G~C)is obtained.For all x,y,z∈{0,1,+,-},the explicit formulas for the reciprocal of Z(λ,G)in terms of r,m,n and the characteristic polynomial of G are obtained.Due to limited space,only the expressions for Gwith z=0,and xyz∈{0++,+++,1+-}are presented here.