Complete characterization of bicyclic graphs with minimal Kirchhoff index

The resistance distance between any two vertices of a graph G is defined as the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index K(G) is the sum of the resistance distances between all the pairs of vertices in G. A bicyclic graph is a connected graph whose number of edges is exactly one more than its number of vertices. In this paper, we completely characterize the bicyclic graphs of order n≥4 with minimal Kirchhoff index and determine bounds on the Kirchhoff index of bicyclic graphs. This improves and extends some earlier results.