The Kirchhoff index of subdivisions of graphs

Let G be a connected graph. The Kirchhoff index (or total effective resistance, effective graph resistance) of G is defined as the sum of resistance distances between all pairs of vertices. Let S ( G ) be the subdivision graph of G . In this note, a formula and bounds for the Kirchhoff index of S ( G ) are obtained. It turns out that the Kirchhoff index of S ( G ) could be expressed in terms of the Kirchhoff index, the multiplicative degree-Kirchhoff index, the additive degree-Kirchhoff index, the number of vertices, and the number of edges of G . Our result generalizes the previous result on the Kirchhoff index of subdivisions of regular graphs obtained by Gao et al. (2012).