Generalization for Laplacian energy

Abstract

Let G be a simple graph with n vertices and m edges. Let λ ₁, λ ₂, ..., λ _n , be the adjacency spectrum of G, and let μ ₁, μ ₂, ..., μ _n be the Laplacian spectrum of G. The energy of G is $E\left( G \right) = \sum\limits_{i = 1}^n {\left| {\lambda _i } \right|}$, while the Laplacian energy of G is defined as $LE\left( G \right) = \sum\limits_{i = 1}^n {\left| {\mu _i - \frac{{2m}} {n}} \right|}$. Let γ ₁, γ ₂, ..., γ _n be the eigenvalues of Hermite matrix A. The energy of Hermite matrix as $HE\left( A \right) = \sum\limits_{i = 1}^n {\left| {\gamma _i - \frac{{tr\left( A \right)}} {n}} \right|}$ is defined and investigated in this paper. It is a natural generalization of E(G) and LE(G). Thus all properties about energy in unity can be handled by HE(A).