Malaysian Journal of Mathematical Sciences, June 2025, Vol. 19, No. 2
The Sombor Index of a Power Graph for Some Finite Groups and Their Sombor Polynomial
Alimon, N. I., Sarmin, N. H., Khasraw, S. M. S., Najmuddin, N., and Semil @ Ismail, G.
Corresponding Email: idayualimon@uitm.edu.my
Received date: 11 May 2024
Accepted date: 4 November 2024
Abstract:
Sombor index is a new geometric background of graph invariants and is also called a valency-based topological descriptor. It is computed by taking the radical of the sum of the squared degrees of two adjacent vertices within a graph. The Sombor polynomial also involves the degrees of two adjacent vertices where its first order derivative at $x$ is one, is equal to the Sombor index. Meanwhile, in a power graph, two different vertices are connected by an edge if and only if one is the power of the other. The graph's vertex set consists of all the elements in a group. In this study, the Sombor index and Sombor polynomial of the power graph for some finite non-abelian groups are determined by using their definitions. The dihedral, generalized quaternion, and quasi-dihedral groups are considered. The generalization of the power graph for the quasi-dihedral groups are also found.
Keywords: Sombor index; Sombor polynomial; power graph; graph theory; group theory
