On the relationship between the Kirchhoff and the Narumi-Katayama indices

Abstract

Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees ∆ = d 1 ≥ d 2 ≥ • • • ≥ d n = δ > 0 and diagonal matrix D = diag(d 1 , d 2 ,. .. , d n) of its vertex degrees. Denote by K f (G) = n n−1 i=1 1 µ i , where µ i are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = n i=1 d i the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices.