For a simple connected graph G of order n, having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, the Laplacian–energy–like invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = ∑n−1 i=1 √ μi and Kf(G) = n ∑n−1 i=1 1 μi , respectively. In this paper, LEL and Kf are compared, and sufficient conditions for the inequality Kf(G) < LEL(G) are established.

for a simple digraph $g$ of order $n$ with vertex set${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote theout-degree and in-degree of a vertex $v_i$ in $g$, respectively. let$d^+(g)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and$d^-(g)=diag(d_1^-,d_2^-,ldots,d_n^-)$. in this paper we introduce$widetilde{sl}(g)=widetilde{d}(g)-s(g)$ to be a new kind of skewlaplacian matrix of $g$, where $widetilde{d}(g...

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...