On the Lanzhou index of graphs

Abstract

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. The Lanzhou index of a graph $G$ is defined as ${\rm Lz}(G)=\sum_{u\in V(G)}\,d_{\overline{G}}(v)\,d_G(v)^2$, where $d_G(v)$ denotes the degree of the vertex $v$ in $G$. In this paper, we determine extremal values of the Lanzhou index in terms of some graph parameters, as well as Nordhaus--Gaddum--type results. We also find relations between Lanzhou Index and other topological indices.