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

Abstract

Let G be a graph. Denote by d u , the degree of a vertex u of G and represent by v w , 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 u − 1 over all edges u v 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.