Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiquesVol. CXLIV, No. 37, pp. 59–70 (2012)
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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 Laplacianûenergyûlike invariantA. Arsic, I. Gutman, K. CH. Das and K. XuFaculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, SerbiaDepartment 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, Laplacianûenergyûlike invariant, LEL Classification (MSC2000): 05C50 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 8 Apr 2013. This page was last modified: 9 Apr 2013.
© 2013 Mathematical Institute of the Serbian Academy of Science and Arts
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