Maximum Modified Sombor Index of Unicyclic Graphs with Given Girth

Abstract

‎For any graph $G$‎, ‎the modified Sombor index is defined as the reciprocal of the well-known Sombor Index‎. ‎The girth of $G$‎, ‎by short $g(G)$‎, ‎is the length of the smallest cycle in $G$‎. ‎A graph with exactly one cycle is a unicyclic graph‎. ‎If it is further‎, ‎connected‎, ‎it is a connected unicyclic graph‎. ‎In this article‎, ‎we achieved the modified Sombor index for a collection of graphs‎, ‎connected unicyclic graphs and also obtained the maximum modified Sombor index for the class of unicyclic graphs based on the restrictions by a fixed girth‎.