Analytical and computational properties of the variable symmetric division deg index

Abstract

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $SD{D}_{\alpha }(G)={\sum }_{uv\in E(G)}(d_u^{\alpha }/d_v^{\alpha }+d_v^{\alpha }/d_u^{\alpha })$, and to characterize graphs extremal with respect to them. Here, $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha \in \mathbb{R}$. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $SD{D}_{\alpha }(G)$ index on random graphs and show that the ratio $⟨SD{D}_{\alpha }(G)⟩/n$ ($n$ being the order of the graph) depends only on the average degree $⟨d⟩$.

16 pages, 1 figure