Kirchhoff index of linear hexagonal chains

Abstract

Abstract The resistance distance r ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf ( G ) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L n consists of the Laplacian spectrum of path P 2 n +1 and eigenvalues of a symmetric tridiagonal matrix of order 2 n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L n is approximately to one half of its Wiener index. Finally, we show that ${Kf(G)\over W(G)}>{1\over 5}$ holds for all graphs G in a class of graphs including L n . © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008