On Laplacian like energy of trees

TL;DR

This paper introduces the Laplacian-like energy of graphs, corrects previous proofs related to graph ordering based on Laplacian coefficients, and generalizes conditions for graph comparisons.

Contribution

It corrects an error in earlier proofs and extends the theoretical framework for comparing graphs using Laplacian coefficients and Laplacian-like energy.

Findings

Identified and corrected an error in previous proof regarding Laplacian coefficient ordering.

Derived the inverse of the Jacobian matrix for derivatives of symmetric polynomials.

Established a generalized theorem with necessary conditions for graph partial ordering.

Abstract

Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed graph invariant, defined as the sum of square roots of Laplacian eigenvalues. For bipartite graphs, the Laplacian--like energy coincides with the recently defined incidence energy $IE (G)$ of a graph. In [D. Stevanovi\' c, \textit{Laplacian--like energy of trees}, MATCH Commun. Math. Comput. Chem. 61 (2009), 407--417.] the author introduced a partial ordering of graphs based on Laplacian coefficients. We point out that original proof was incorrect and illustrate the error on the example using Laplacian Estrada index. Furthermore, we found the inverse of Jacobian matrix with elements representing derivatives of symmetric polynomials of order $n$, and provide a corrected elementary proof of the fact: Let $G$ and $H$ be two $n$-vertex graphs; if for Laplacian coefficients holds $c_k (G) \leqslant c_k (H)$ for $k = 1, 2, ..., n - 1$, then $LEL (G) \leqslant LEL (H)$. In addition, we generalize this theorem and provide a necessary condition for functions that satisfy partial ordering based on Laplacian coefficients.