Bicyclic Graphs With Extremal Kirchhoff Index

A connected graph G is viewed as an electrical network N by replacing each edge of G with a unit resistor. In N, the resistance distance between any two vertices is the effective resistance between them. Kirchhoff index of G is defined to be the sum of resistance distance between all pairs of vertices in N. A bicyclic graph is a connected graph whose edge number is one more than its vertex number. Let S_n^p,q denote the graph obtained from cycles C_p and C_q by attaching n + l-p-q pendent edges to the unique common vertex of them. Let P_n^p,q be the graph consisting of two disjoint cycles C_p and C_q and a path of length n-p-q+1 joining them (when n-p-q+1 =0, P_n^p,q coincides with S_n^p,q. In this paper, we show that among all n-vertex bicyclic graphs with exactly two cycles: (a) P_n^3,3 has the maximal Kirchhoff index, (b) the following graphs have the minimal Kirchhoff indices: S₅^3,3; S₆^3,4; S₇^4,4; S₅^4,4 and S₈^4,5; S_n^4,4 (9≤n≤11); S₁₂^4,4, S₁₂^3,4 and S₁₂^3,3; S_n^3,3 (n > 12).