Normalized Laplacian Energy and Distance Based Energy for Cross Monic Zero Divisor Graphs Associated with Commutative Ring

Cross monic zero divisor graph for a commutative ring R is a connected graph, denoted by CM ZG ( Z(n) x Z(m)[x]/(f(x))) with order xi, whose vertices are non-zero zero divisors Z(R)/{0} of commutative ring, and two vertices u, v are connected by an edge if and only if uv = 0. In this paper, we discuss energy, Laplacian energy, distance energy and distance signless Laplacian energy for CM ZG (Z(2) x Z(p>2)[x] /(x(2))) and CM ZG (Z(p) x Z(p)[x]/(x(2))) . Also, we determine the normalized Laplacian energy.