Kirchhoff Index In Weighted Graphs

The resistance distance r_ij between vertices i and j of a weighted connected graph G is computed as the effective resistance between nodes i and j in the corresponding network N, which is constructed from G by replacing each edge of G with a fixed resistor, and the weight assigned on each edge is equal to the value of the corresponding resistor. Analogue to the Wiener index, Klein and Randic defined the Kirchhoff index Kf(G) of G, which is the sum of resistance distances between all pairs of vertices in G. In this work, firstly, according to the generalized inverse theory of singular matrix we demonstrate that the formulae for Kirchhoff index in unweighted graphs which is derived in terms of Laplacian spectrum also holds for weighted graphs, which extends the conclusion found by Klein and Gutman. Secondly, using the formulae mentioned above and circulant matrix theory, we obtain the Kirchhoff index of weighted wheel graph W_n(a, b) as well as the asymptotic behavior of Kf(W_n(a, b)):Finally, a new formula for Kirchhoff index of weighted graphs is derived in terms of the coefficients of the Laplacian eigenpolynomial, by which we obtain the Kirchhoff index of some special classes of weighted graphs.