Geodesic
Relations between Kirchhoff index and Laplacian–energy–like invariant
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 37 (2012) no. 1
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The Kirchhoff index Kf and the Laplacian–energy– like invariant LEL are two graph invariants defined in terms of the Laplacian eigenvalues. If $\mu_1 \ge\mu_2 \ge...\mu_{n-1} > \mu_n=0$ are the Laplacian eigenvalues of a connected n-vertex graph, then $ Kf = \sum{i=1}{n-1}\frac{1}{\mu_i}$ and $LEL = um{i=1}{n-1}qrt{\mu_i}$. 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. 
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     author = {B. Arsi\'c and I. Gutman and K. Ch. Das and K. Xu},
     title = {Relations between {Kirchhoff} index and {Laplacian{\textendash}energy{\textendash}like} invariant},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {59 - 70},
     year = {2012},
     volume = {37},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/BASS_2012_37_1_a3/}
}
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B. Arsić; I. Gutman; K. Ch. Das; K. Xu. Relations between Kirchhoff index and Laplacian–energy–like invariant. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 37 (2012) no. 1. http://geodesic.mathdoc.fr/item/BASS_2012_37_1_a3/