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) =∑n i=1|µi − d|, where µi is the i-th eigenvalue of Laplacian matrix of G, and d is average degree. Similarly, the Quasi-Laplacian energy of the graph G of order n is defined as EQ (G) =∑n i=1σ2i, whereσi is the i-th eigenvalue of signless Laplacian matrix of G. The inequality LE(Pn ) ≤ LE(Tn ) ≤ LE(Sn ) was given as conjecture by Radenković and Gutman in 2007. This conjecture was proved by Trevisan et al. for trees of diameter 3 and by Rehman et al. for trees of diameter 4. 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 EQ (Pn ) ≤ EQ (Tn ) ≤ EQ (Sn ) inequality, which is similar to Gutman’s conjecture for the Quasi-Laplacian energy.