Laplacian energy and Quasi-Laplacian energy for trees

For a simple graph $G$, the Laplacian matrix is denoted by $L(G)$ and the Signless Laplacian matrix is denoted by $Q(G)$. These matrices and their eigenvalues are widely used in many fields such as network measurement and chemical structures. Laplacian energy of a graph $G$ of order $n$ is defined as $LE(G)= \sum_{i=1}^n \ |\mu_i-\overline d |$, where $\mu_i$ is the $i$-th eigenvalue of Laplacian matrix of $G$, and $\overline d$ is average degree. Similarly, the Quasi-Laplacian energy of the graph $G$ of order $n$ is defined as $E_Q(G)= \sum_{i=1}^n \ \sigma_i^2$, where $\sigma_i$ is the $i$-th eigenvalue of Signless Laplacian matrix of $G$.The inequality $LE(P_n) \leq LE(T_n) \leq LE(S_n)$ was given as conjecture by Radenković and Gutman in 2007. This conjucture was proved by Trevisan et al. for trees with 3 diameters and by Rehman et al. for trees with 4 diameters. Our aim is to prove this conjecture for trees with larger diameters. In this study, we solved the above conjecture for some tree classes between 5 and 15 in diameter by the linear time algorithm for characteristic polynomials. It is also proved ${E_Q}\left( {{P_n}} \right) \le {E_Q}\left( {{T_n}} \right) \le {E_Q}\left( {{S_n}} \right) $ inequality, which is similar to Gutman's conjecture for the Quasi-Laplacian energy.