On some lower bounds for the Kirchhoff index

Abstract

Let G = (V;E), V = f1;2;:::;ng, be a simple connected graph of order n and size m, with sequence of vertex degrees ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(i). Denote by m1 ≥ m2 ≥ ··· ≥ mn-1 > mn = 0 the Laplacian eigenvalues of G. Further, denote with K f(G) = n∑n i=-11 m1i and t = t(G) = 1 n ∏n i=-11 mi, the Kirchhoff index and the number of spanning trees of G, respectively. In this paper we determine several lower bounds for K f(G) depending on t(G) and some of the graph parameters n, m or ∆.