Double Hexagonal Chains with the Extremal \(PI\) Indices

Abstract

The Padmakar-Ivan (\(PI\)) index of a graph \(G = (V, E)\) is defined
as \(PI(G) = \sum_{e \in uv} (n_{eu}(e|G) + n_{ev}(e|G))\)
where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\)
than to \(v\) and \(n_{ev}(e|G)\) is the number of edges of \(G\) lying
closer to \(v\) than to \(u\).

In this paper, we derive a recursive formula for computing the
\(PI\) index of a double hexagonal chain using the orthogonal cut,
and characterize the double hexagonal chains with extremal
\(PI\) indices.