EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques
Vol. CXLIV, No. 37, pp. 59–70 (2012)

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Relations between Kirchhoff index and Laplacianűenergyűlike invariant

A. Arsic, I. Gutman, K. CH. Das and K. Xu

Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, P. R. China

Abstract: The Kirchhoff index Kf and the Laplacian.energy. like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If $\mu_1\geq \mu_2 \geq \cdots \geq \mu_{n-1}>\mu_n = 0$ are the Laplacian eigenvalues of a connected n-vertex graph, then $K f=n\sum_{i=1}^{n-1} 1/\mu_i$ and $LEL=\sum_{i=1}^{n-1}\sqrt{\mu_i}.$ We examine the conditions under which $Kf > LEL.$ Among other results we show that $Kf > LEL$ holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a finite number of graphs. These exceptional graphs are determined.

Keywords: Laplacian spectrum (of graph), Laplacian eigenvalue, Kirchhoff index, Laplacianűenergyűlike invariant, LEL

Classification (MSC2000): 05C50

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Electronic fulltext finalized on: 8 Apr 2013. This page was last modified: 9 Apr 2013.

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