The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

The power graph is a special type undirected simple graph of a finite group G which has the group elements as its vertex set and two distinct vertices [Formula: see text] are adjacent if one is non-negative integral power of other. Vertex labeling of graph [Formula: see text] is a process of assigning integers to all its vertices subject to certain conditions. In simple words, vertex (edge) labeling is a function of all vertices (edges) to set of labels. Frequently, integers are used in labeling of vertices and edges. A graph [Formula: see text] is said to be divisor graph if all vertices of graph can be labeled with positive integers such that any two distinct vertices [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. In this research paper, we prove that every power graph of finite group is always a divisor graph, but converse is not true.

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The co-prime order graph Θ(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 Θ(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 Θ(G) when G is a finite Abelian p-group, Z p t × Z q s or a dihedral group D p n .

In this paper, (1) we deduce a formula for the number of automorphism of finite Abelian p-group of rank two ( )