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Suppose is a finite group and is a subset of. The commuting graph on the set , whose vertex set with any two vertices connected by an edge if and only if they commute. In this paper, we consider as the Mathieu groups, symplectic groups, together with their automorphism groups and are conjugacy classes of involutions. Let. Here we investigate the orbits under the action of from a fixed vertex and describe the group theoretic structure of where is a-orbit representative.

A commuting graph is a graph denoted by C(G,X) where G is any group and X, a subset of a group G, is a set of vertices for C(G,X) . Two distinct vertices, x,y∈X , will be connected by an edge if the commutativity property is satisfied or xy=yx . This study presents results for the connectivity of C(G,X) when G is a symmetric group of degree n, Sym (n) , and X is a conjugacy class of elements of order three in G.

The atom-bond connectivity (ABC) index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G) = ∑ uv∈E(G) √[d v + du – 2]/[d v · d u ], where du denotes the degree of a vertex u in G. In this paper, we establish the general formulas for the atom bond connectivity index of molecular graphs of alkenes and cycloalkenes.

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.