The Zig-Zag Chain as an Extremal Value of VDB Topological Indices of Polyomino Chains

Abstract

We give conditions on the numbers $\{{\phi }_{ij}\}$ under which a vertex-degree-based topological index $TI$ of the form

$$TI(G)=\sum _{1\le i\le j\le n-1}{m}_{ij}{\phi }_{ij},$$

where $G$ is a graph with $n$ vertices and $m_{ij}$ is the number of $ij$-edges, has the zigzag chain as an extreme value among all polyomino chains. As a consequence, we deduce that over the polyomino chains, the zigzag chain has the maximal value of the Randić index, the sum-connectivity index, the harmonic index, and the geometric-arithmetic index, and the minimal value of the first Zagreb index, second Zagreb index, and atom-bond-connectivity index.