On Laplacian like energy of trees (Q2853192)

The Laplacian-like energy of a graph \(G\), denoted by \(\mathrm{LEL}(G)\), is newly proposed graph invariant, defined as the sum of square roots of Laplacian eigenvalues. The authors prove the following result. Let \(G\) and \(H\) be two \(n\)-vertex graphs. If for Laplacian coefficients \(c_k(G)\leq c_k(H)\) holds for \(k=1,2,\dots,n-1\), then \(\mathrm{LEL}(G)\leq\mathrm{LEL}(H)\). Furthermore, this result is generalized and a necessary condition for functions that satisfy a partial ordering based on Lapalacian coefficients is provided.