The Kirchhoff index of subdivisions of graphs

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