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 Φ Q of a tree digraph Q with n vertices, e r e ( Φ Q ) ≤ min n ( 2 a + 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 = 1 n [ 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. [ABSTRACT FROM AUTHOR]