Computing the Kirchhoff Index of Some xyz-Transformations of Regular Molecular Graphs

摘要

Let G be a connected molecular graph. The resistance distance between any two vertices of G is defined as the effective resistance between the two corresponding nodes in the electrical network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index of G is defined as the sum of resistance distances between all pairs of vertices. Gao et al. (2012) and You et al. (2013) gave formulae for the Kirchhoff index of two types of xyz - transformations, namely, the subdivision graph and the total graph, of regular graphs. In this paper, we compute the Kirchhoff index of some other xyz - transformations of regular (molecular) graphs, with explicit formulae for the Kirchhoff index of these transformation graphs being given in terms of parameters of the original graph.