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

We give conditions on the numbers \(\{\varphi_{ij}\}\) under which a vertex-degree-based topological index \(TI\) of the form

\[
TI(G) = \sum_{1\leq i\leq j\leq n-1} m_{ij}\varphi_{ij},
\]

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.