Prime Graph Over Semidihedral Group and Its Connectivity Indices

Abstract

This study investigates the structural properties and topological connectivity indices of the prime graph associated with the semidihedral group of order 2 raised to the power of n, whose vertex set consists of all elements of the group. Two distinct vertices are adjacent precisely when the greatest common divisor of their orders is a prime number. Since the identity element has order one, it is not adjacent to any other vertex and therefore forms an isolated vertex. Consequently, the prime graph is disconnected and consists of the isolated identity element together with a non-trivial connected component containing all remaining group elements. The vertex set is partitioned into three disjoint subsets consisting of the identity element, elements of order two, and elements of order greater than two. Based on this partition, we characterize the degree sequence, diameter, radius, and clique number of the graph. Furthermore, exact closed-form analytical formulas are obtained for several fundamental degree-based and distance-based topological connectivity indices, including the First Zagreb index, Wiener index, Hyper-Wiener index, Harary index, and Forgotten index.