Extremal values of vertex-degree-based topological indices over graphs

Abstract

Given a graph $G$ with $n$ vertices, a vertex-degree-based topological index is defined from a set of real numbers $\left\{ \varphi _{ij}\right\}$ as $TI\left( G\right) =\sum m_{ij}\left( G\right) \varphi _{ij}$, where $m_{ij}\left( G\right)$ is the number of edges between vertices of degree $i$ and degree $j$, and the sum runs over all $1\le i\le j\le n-1$. In this paper we show that under certain conditions on the associated function $\widehat{f}\left( i,j\right) =\frac{ij\varphi _{ij}}{i+j}$ and its partial derivatives, the extremal values of $\widehat{f}$ are attained in the three points $\left( 1,1\right) ,\left( 1,n-1\right)$ and $\left( n-1,n-1\right)$ of the region $\widehat{K}=\left\{ \left( i,j\right) \in {\mathbb {R}}\times {\mathbb {R}} :1\le i\le j\le n-1\right\}$, each of these points corresponding to $K_{2}\cup K_{2}\cup \cdots \cup K_{2}$ (or $K_{2}\cup K_{2}\cup \cdots \cup K_{2}\cup P_{3}$ if $n$ is odd), the star $S_{n}$ and the complete graph $K_{n}$, respectively. As an application of this result, we find the extremal values of the well-known vertex-degree-based topological indices over ${\mathcal {G}}_{n}$, the set of graphs with $n$ non-isolated vertices.