Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Abstract

Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as L E = ∑ i = 1 n | μ i - 2 m n | LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|} , L E L = ∑ i = 1 n - 1 μ i LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} } and K f = n ∑ i = 1 n - 1 1 μ i Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.