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SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 04 October 2017, pp. 1-10
- Print publication:
- February 2018
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- Article
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- You have access
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Let
$G$ be a simple connected graph with 
$n$ vertices and 
$m$ edges and 
$d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$ its sequence of vertex degrees. If 
$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$ are the Laplacian eigenvalues of 
$G$, then the Kirchhoff index of 
$G$ is 
$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for 
$\mathit{Kf}(G)$ in terms of some of the parameters 
$\unicode[STIX]{x1D6E5}=d_{1}$, 
$\unicode[STIX]{x1D6E5}_{2}=d_{2}$, 
$\unicode[STIX]{x1D6E5}_{3}=d_{3}$, 
$\unicode[STIX]{x1D6FF}=d_{n}$, 
$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$ and the topological index 
$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.
