The co-prime order graph \(\Theta (G)\) of a given finite group is a simple undirected graph whose vertex set is the group \(G\) itself, and any two vertexes \(x,y\) in \(\Theta (G)\) are adjacent if and only if \(gcd(o(x),o(y))=1\) or prime. In this paper, we derive a precise formula to count the vertex's degree in the co-prime order graph of a finite Abelian group or dihedral group.We also investigate the Laplacian spectrum of the co-prime order graph \(\Theta (G)\) when G is a finite Abelian p-group, \({\mathbb{Z}_p}^t \times {\mathbb{Z}_q}^s\) or a dihedral group \(D_{p^n}\).

The cubic power graph of finite group G with identity element e, is an undirected finite, simple graph in which a pair of distinct vertices x, y have an edge iff xy = z3 or yx = z3 for any z ∈ Dn with z3 ≠ e. In this paper, we have studied the structural representation of the cubic power graph of the dihedral group and various structural properties such as clique, girth, vertex degree, chromatic number, independent number, matching number, perfect matching, dominating number, etc. We have also calculated various topological indices such as the Harary index, the first and second Zagreb indices, the Wiener and hyper-Wiener indices, the Schultz index, the harmonic index, the general Randic index, the eccentric connectivity index, the Gutman index, the atomic-bond connectivity index, and the geometricarithmetic index of the cubic power graph of dihedral group Dn when gcd(n, 3) = 1.

In this paper, (1) we deduce a formula for the number of automorphism of finite Abelian p-group of rank two and (2) a formula for the number of auto-morphism of finite Abelian group of rank two Z_m×Z_n, by using partition of number of cyclic subgroups of group .

We denote by D(Z_p[x, 2]), the divisor graph of set of all polynomials of degree at most two whose coefficients are from field Z_p as its vertex set and any two distinct vertices in it are adjacent if one is a divisor of the other. In the present paper, we determine the degree of each vertex of the divisor graph D(Z_p[x, 2]) and also study its characteristics as complete and/or cyclic or not. We also determine girth, size, degree sequence and irregularity index for the graph D(Z_p[x, 2]). We also derive necessary and sufficient condition for D(Z_p[x, 2]) to be Hamiltonian and/or Eulerian.