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Bounds for the Kirchhoff index via majorization techniques

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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

$$K(G) \geq \frac{n}{d_{1}} \left[ \frac{1}{1+\frac{\sigma}{\sqrt{n-1}}} + \frac{(n-2)^{2}}{n-1-\frac{\sigma}{\sqrt{n-1}}}\right] ,$$

where d 1 is the largest degree among all vertices in G,

$$\sigma ^{2} = \frac{2}{n} \sum_{(i, j) \in E} \frac{1}{d_{i}d_{j}} = \left( \frac{2}{n}\right) R_{-1}(G),$$

and R −1(G) is the general Randić index of G for \({\alpha =-1}\) . Also we show that

$$K(G) \leq \frac{n}{d_{n}}\left( \frac{n-k-2}{1-\lambda _{2}}+\frac{k}{2}+\frac{1}{\theta}\right) ,$$

where d n is the smallest degree, \({\lambda _{2}}\) is the second eigenvalue of the transition probability of the random walk on G,

$$k = \left \lfloor \frac{\lambda _{2} \left( n-1\right) +1}{\lambda _{2}+1}\right\rfloor {\rm and}\quad\theta = \lambda _{2} \left( n-k-2\right) -k+2.$$

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Correspondence to José Luis Palacios.

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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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