Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Let G = (V, E), V = {v1 , v2, . . ., vn}, be a simple connected graph of order n and size m. Denote with μ1 ≥ μ2 ≥ · · · ≥ μn−1 > μn = 0 eigenvalues of the Laplacian matrix L(G) of G. The Kirchhoff index and the number of spanning trees of G expressed in terms of Laplacian eigenvalues are given by Kf (G) = n Pni=1−1 1 µi and t(G) = n1 Qni=1−1 μi, respectively. The characteristic polynomial of L(G) is given by φ(L(G)) = Pnk=0 pkxn−k. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of G. Then, we use it to improve the previously obtained results on Kf(G) and t(G). In addition, we present new Nordhaus–Gaddumtype inequalities for Kf(G) and t(G).