Coxeter energy of graphs

We study the concept of the Coxeter energy of graphs and digraphs (quivers) as an analogue of Gutman's adjacency energy, which has applications in theoretical chemistry and is a recently widely investigated graph invariant. Coxeter energy CE(G)CE(G) of a (di)graph G is defined to be the sum of the absolute values of all complex eigenvalues of the Coxeter matrix associated with G. Our main inspiration for the study comes from the Coxeter formalism appearing in group theory, Lie theory, representation theory of algebras, mathematical physics and other contexts. We focus on the Coxeter energy of trees and we prove that the path (resp. the maximal star) has the smallest (resp. the greatest) Coxeter energy among all trees (resp. two large subclasses of trees) with fixed number of vertices. We provide several other related results, as the characterization of trees with second smallest and second greatest Coxeter energy, bounds for Coxeter energy, and general facts on Coxeter spectra of graphs extending known results e.g. for Salem trees and for certain special real Coxeter eigenvalues of trees. Additionally, we discuss few other energy-like quantities for the Coxeter spectra of (di)graphs, including Coxeter energy “normalized” by the trace of the Coxeter matrix and the quantities derived from variants of Coulson integral formula.