A note on the Kirchhoff index of graphs

Let G be a simple connected graph with n vertices, m edges, and a sequence of vertex degrees Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ > 0 . Denote by μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n − 1 > μ n = 0 the Laplacian eigenvalues of G . The Kirchhoff index of G is defined as K f ( G ) = n ∑ n − 1 i = 1 1 μ i . A couple of new lower bounds for K f ( G ) that depend on n , m , Δ and some other graph invariants are obtained.