Extremal Gourava indices of unicyclic graphs

Abstract Topological indices are useful molecular descriptors to measure Quantitative Structure-Activity Relationship (QSAR), Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Toxicity Relationship (QSTR). Recently, some novel topological indices of graphs, called the first Gourava index ($CO_1$) and the second Gourava index ($CO_2$), have been proposed to characterize the nature of chemical compounds or interconnection networks. The first and the second Gourava indices of the graph $G$ are denoted by $C{O_1}(G) =\sum\limits_{uv \in E(G)} {[{d_u}} + {d_v} + {d_u}{d_v}]$ and $C{O_2}(G) = \sum\limits_{uv \in E(G)} {[({d_u}} + {d_v}){d_u}{d_v}]$, where $d_u$ is the degree of vertex $u$. In this work, we investigate the extremal values on the first and the second Gourava indices of $n$-vertex unicyclic graphs and characterize the unicyclic graphs that achieve the extremes. We show that, for $n$-vertex unicyclic graph $G$, $8n \le C{O_1}(G) \le 2{n^2} - n + 9$ and $16n \le C{O_2}(G) \le n^3 +3n + 12$, where the lower bound is achieved by $C_n$ and the upper bound is achieved by $G_{3,1}^{(n)}$, which is obtained by attaching $n- 3$ leaves to one vertex of $C_3$. Furthermore, we investigate the applications of Gourava indices to the benzenoid hydrocarbons and show that they can predict the physico-chemical properties of molecules precisely.