Minimizing the Laplacian-energy-like of graphs

Let G be a connected simple graph with order n and Laplacian matrix $L\left(G\right)$. The Laplacian-energy-like of G is defined as$LEL\left(G\right)=\sum _{i=1}^n\sqrt{{\lambda }_i},$ where ${\lambda }_i$ is the eigenvalue of $L\left(G\right)$ for $i=1,\dots ,n$. In this paper, with the aid of Ferrers diagrams of threshold graphs, we provide an algebraic combinatorial approach to determine the graphs with minimal Laplacian-energy-like among all connected graphs having n vertices and m edges, showing that the extremal graph is a special threshold graph, named as the quasi-complete graph.