On-Bond Incident Degree Indices of Square-Hexagonal Chains

Abstract

For a graph $G$ , its bond incident degree (BID) index is defined as the sum of the contributions $f\left(d_u,d_v\right)$ over all edges $uv$ of $G$ , where $d_w$ denotes the degree of a vertex $w$ of $G$ and $f$ is a real-valued symmetric function. If $f\left(d_u,d_v\right)=d_u+d_v$ or $d_u{d}_v$ , then the corresponding BID index is known as the first Zagreb index $M_1$ or the second Zagreb index $M_2$ , respectively. The class of square-hexagonal chains is a subclass of the class of molecular graphs of minimum degree 2. (Formal definition of a square-hexagonal chain is given in the Introduction section). The present study is motivated from the paper (C. Xiao, H. Chen, Discrete Math. 339 (2016) 506–510) concerning square-hexagonal chains. In the present paper, a general expression for calculating any BID index of square-hexagonal chains is derived. The chains attaining the maximum or minimum values of $M_1$ and $M_2$ are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.