Unicyclic graphs with extremal Lanzhou index

Abstract

Very recently D. Vukičević et al. [8] introduced a new topological index for a molecular graph G named Lanzhou index as \(Lz\left( G \right) = \sum\nolimits_{u \in V\left( G \right)} {\overline {{d_u}} } d_u^2\), where d_u and \(\overline {{d_u}} \) denote the degree of vertex u in G and in its complement respectively. Lanzhou index Lz(G) can be expressed as (n − 1)M₁(G) − F (G), where M₁(G) and F (G) denote the first Zagreb index and the forgotten index of G respectively, and n is the number of vertices in G. It turns out that Lanzhou index outperforms M₁(G) and F(G) in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers. It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively. In this paper, we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.