On the Maximum Symmetric Division Deg Index of k‐Cyclic Graphs

Abstract

Let G be a graph. Denote by du, the degree of a vertex u of G and represent by vw, the edge of G with the end‐vertices v and w. The sum of the quantities over all edges uv of G is known as the symmetric division deg (SDD) index of G. A connected graph with n vertices and n − 1 + k edges is known as a (connected)k‐cyclic graph. One of the results proved in this study is that the graph possessing the largest SDD index over the class of all connectedk‐cyclic graphs of a fixed order n must have the maximum degree n − 1. By utilizing this result, the graphs attaining the largest SDD index over the aforementioned class of graphs are determined for every k = 0,1, …, 5. Although, the deduced results, for k = 0,1,2, are already known, however, they are proved here in a shorter and an alternative way. Also, the deduced results, for k = 3,4,5, are new, and they provide answers to two open questions posed in the literature.