The Kirchhoff index of subdivisions of graphs
Let G G be a connected graph. The Kirchhoff index (or total effective resistance, effective graph resistance) of G G is defined as the sum of resistance distances between all pairs of vertices. Let S(G) S ( G ) be the subdivision graph of G G . 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 G G . Our result generalizes the previous result on the Kirchhoff index of subdivisions of regular graphs obtained by Gao etal. (2012).
kirchhoff index subdivision total effective resistance resistance distance