Comparison between Kirchhoff index and the Laplacian-energy-like invariant

Abstract Let G be a connected graph of order n with Laplacian eigenvalues μ 1 ⩾ μ 2 ⩾ ⋯ ⩾ μ n - 1 > μ n = 0 . The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as Kf = n ∑ k = 1 n - 1 1 / μ k and LEL = ∑ k = 1 n - 1 μ k , respectively. We compare Kf and LEL and establish two sufficient conditions under which LEL Kf . The connected graphs of order n with nine greatest Kirchhoff indices are determined; for these LEL > Kf holds.

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