Abstract
Let G be a graph. Denote by d(u), 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 [(d(u))(2) + (d(v))(2)][(d(u))(-1)(d(v))(-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 connected k-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.