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 $vw$ , the edge of $G$ with the end-vertices $v$ and $w$ . The sum of the quantities $\left[{\left(d_u\right)}^2+{\left(d_v\right)}^2\right]\left[{\left(d_u\right)}^{-1}{\left(d_v\right)}^{-1}\right]$ 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 $k=\mathrm{0,1},\dots ,5$ . Although, the deduced results, for $k=\mathrm{0,1,2}$ , are already known, however, they are proved here in a shorter and an alternative way. Also, the deduced results, for $k=\mathrm{3,4,5}$ , are new, and they provide answers to two open questions posed in the literature.