Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ℝ, 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 d1 is the largest degree among all vertices in G, and R-1(G) is the general Randić index of G for α =-1. Also we show that, where dn is the smallest degree, λ2 is the second eigenvalue of the transition probability of the random walk on G,.
Torriero, A., Bianchi, M., Cornaro, A., Palacios, J. (2013). Bounds for the Kirchhoff index via majorization techniques. JOURNAL OF MATHEMATICAL CHEMISTRY, 51(2), 569-587 [10.1007/s10910-012-0103-x].
Bounds for the Kirchhoff index via majorization techniques
Torriero, Anna;Bianchi, Monica;Cornaro, Alessandra;
2013
Abstract
Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ℝ, 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 d1 is the largest degree among all vertices in G, and R-1(G) is the general Randić index of G for α =-1. Also we show that, where dn is the smallest degree, λ2 is the second eigenvalue of the transition probability of the random walk on G,.
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