Complete characterization of bicyclic graphs with minimal Kirchhoff index

The resistance distance between any two vertices of a graph GG is defined as the network effective resistance between them if each edge of GG is replaced by a unit resistor. The Kirchhoff index K(G)K(G) is the sum of the resistance distances between all the pairs of vertices in GG. 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≥4n≥4 with minimal Kirchhoff index and determine bounds on the Kirchhoff index of bicyclic graphs. This improves and extends some earlier results.