Minimal hexagonal chains with respect to the Kirchhoff index. (English) Zbl 1497.05046
Summary: Let \(G\) be a connected graph. The resistance distance between any two vertices of \(G\) is equal to the effective resistance between them in the corresponding electrical network constructed from \(G\) by replacing each edge with a unit resistor. The Kirchhoff index of \(G\) is defined as the sum of resistance distances between all pairs of vertices. Hexagonal chains are graph representations of unbranched catacondensed benzenoid hydrocarbons. It was shown in Y. Yang and D. J. Klein [Discrete Appl. Math. 175, 87–93 (2014; Zbl 1297.05078)] that among all hexagonal chains with \(n\) hexagons, the linear chain \(L_n\) is the unique chain with maximum Kirchhoff index. However, for hexagonal chains with minimum Kirchhoff index, it was only claimed that the minimum Kirchhoff index is attained only when the hexagonal chain is an “all-kink” chain. In this paper, by standard techniques of electrical networks and comparison results on Kirchhoff indices of \(S\), \(T\)-isomers, “all-kink” chains with maximum and minimum Kirchhoff indices are characterized. As a consequence, hexagonal chains with minimum Kirchhoff indices are singled out.
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C92 | Chemical graph theory |
| 05C12 | Distance in graphs |
| 05C40 | Connectivity |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
Keywords:
resistance distance; Kirchhoff index; “all-kink” chain; \(\Delta - Y\) transformation; \(S,T\)-isomersCitations:
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