RELATIONS BETWEEN KIRCHHOFF INDEX AND LAPLACIAN–ENERGY–LIKE INVARIANT


B. ARSIC, I. GUTMAN, K. CH. DAS, K. XU




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.