Bounds for the Kirchhoff index via majorization techniques

摘要

Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ?, we find upper and lower bounds for the Kirchhoff index K(G) of an arbitrary simple connected graph G that improve those existing in the literature. Specifically we show that, where d_1 is the largest degree among all vertices in G, and R_(-1)(G) is the general Randi? index of G for α =-1. Also we show that, where d_n is the smallest degree, λ_2 is the second eigenvalue of the transition probability of the random walk on G,.