On level energy and level characteristic polynomial of rooted trees

Based on the level index, a Wiener-like topological index proposed by Balaji and Mahmoud [J.Appl.Probab.54 (2017), 701-709], we define the level matrix and study the level energy and the level characteristic polynomial of rooted trees.We establish relations between the level matrix and the usual distance matrix.We also determine various bounds on the level energy and calculate the level energy for specific tree families.Moreover, we provide an explicit expression of the level characteristic polynomial of the so-called rooted double stars and rooted binary caterpillars.Finally, we propose (and provide evidence to support) a conjecture that the rooted path maximises the level energy among all trees with a given number of vertices.