Bounds related to Coxeter spectral measures of graphs

Let M be a real square matrix. We give upper bounds for the sum of the absolute values of the (real part of the) eigenvalues of M, quantity in some particular cases known as (real) energy of M. From these results we obtain a combinatorial bound for the real energy of the Coxeter matrix ΦQ of a tree digraph Q with n vertices,ere(ΦQ)≤minn(2a+b+c+d)2, where a is the number of edges, b and c are respectively the number of bifurcation and congregation paths of Q (as defined below), d=∑i=1n[δ(i)−1][δ(i)−2] with δ(i) the degree of a vertex i, and where the minimum is taken over all possible orientations of edges in Q. As particular case we consider Dynkin, Euclidean and star graphs, obtaining practical bounds for the real Coxeter energy of these classes of digraphs.