Journal of Mathematical Chemistry

, Volume 51, Issue 2, pp 569–587 | Cite as

Bounds for the Kirchhoff index via majorization techniques

  • Monica Bianchi
  • Alessandra Cornaro
  • José Luis Palacios
  • Anna Torriero
Original Paper

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 d1 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 dn is the smallest degree, \({\lambda _{2}}\) is the second eigenvalue of the transition probability of the random walk on G,
$$k = \left \lfloor \frac{\lambda _{2} \left( n-1\right) +1}{\lambda _{2}+1}\right\rfloor {\rm and}\quad\theta = \lambda _{2} \left( n-k-2\right) -k+2.$$

Keywords

Majorization Schur-convex functions Graphs Kirchhoff index 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Monica Bianchi
    • 1
  • Alessandra Cornaro
    • 1
  • José Luis Palacios
    • 2
  • Anna Torriero
    • 1
  1. 1.Department of Mathematics and EconometricsCatholic UniversityMilanItaly
  2. 2.Department of Scientific Computing and StatisticsSimón Bolívar UniversityCaracasVenezuela

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