Abstract
Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of \({\mathbb{R}^{n}}\) , 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 Randić index of G for \({\alpha =-1}\) . Also we show that
where d n is the smallest degree, \({\lambda _{2}}\) is the second eigenvalue of the transition probability of the random walk on G,
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Bianchi, M., Cornaro, A., Palacios, J.L. et al. Bounds for the Kirchhoff index via majorization techniques. J Math Chem 51, 569–587 (2013). https://doi.org/10.1007/s10910-012-0103-x
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DOI: https://doi.org/10.1007/s10910-012-0103-x