The Szeged Index Energy of Commuting Graph of Certain Finite Non-commutative Groups

Let G be a finite group having center Z(G). The commuting graph of G denoted by C(G) has vertex set as G \ Z(G), and two vertices x and y are adjacent in C(G) if x and y commute with each other. The commuting graph of a finite group is a powerful tool in group theory to understand the internal structure of the group. Through its graph-theoretic properties, the commuting graph helps in classifying groups and understanding how the group works internally. It serves as a visual and computational tool to complement other algebraic methods in group theory. The Szeged index is a topological index used in the study of molecular structures, particularly in chemistry and chemical graph theory. It is a numerical value that characterizes the connectivity of a molecular graph. In this paper, we have determined the Szeged index of the commuting graph of various finite non-commutative groups, namely the dihedral group Dn, and the dicyclic group Dicn. Moreover, we determine the energy of C(Dn) and C(Dicn). A graph is said to be hyperenergetic if the energy of G is greater than the complete graph. In this paper, we prove that the graphs C(Dn) and C(Dicn) are non-hyperenergetic graphs.