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.

Let G be a finite abelian group with identity e. The additive power graph Γ_apg(G) of G is the simple undirected graph whose vertex set is group G itself and two distinct vertices a and b of Γ_apg(G) are adjacent if and only if a + b = nc for some c ∈ G with nc ≠ e where n ≥ 2. When n = 2, the additive power graph has been studied in the name of square power graph of finite abelian groups denoted by Γ_sq(G). In this paper we study the the structure and connectedness of square power graph of finite abelian group specifically for Zn and $Z3m×Z2n$⁠.

For k ≥ 2, kth-power graph of a finite abelian group G with identity element e, Γk(G) is a graph in which vertex elements set is group G itself and any two dist

For dihedral group Dn of order 2n with identity element R0, Square power graph of dihedral group Dn is an undirected finite, simple graph having pair of distinct vertices X1,X2 have edge iff X1 X2 = X2 or X2 X1 = X2 for any X ∈ Dn where X2 ≠ R0. In this research, we have calculated characteristic polynomials of degree of vertex based matrices such as maximum and minimum matrix of the Square power graph of dihedral group Dn of order 2n with odd natural number n.