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

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 du2+dv2du−1dv−1 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.