On Coxeter Spectral Study of Posets and a Digraph Isomorphism Problem

Following the spectral graph theory and algebraic techniques in graph theory, we continue a Coxeter spectral study of finite posets and edge-bipartite graphs (or signed graphs in the sense of Harary and Zaslavsky). A connection between properties of the Coxeter spectrum and digraph isomorphism problem for Hasse digraphs of positive and non-negative posets J is also studied. In particular, we study in details a class of posets J with a non-negativity condition, in connection with the Coxeter spectral properties of the simply-laced Euclidean diagrams. We show that symbolic and numerical computer calculations in Python, C and Linux tools allow us to present a complete classification of these posets J, with at most 15 points, by means of their Coxeter spectra. The main classification ideas and the algorithms used in the classification are presented in Sections 4 and 6. We end the paper by showing how our poset classification results apply to the isomorphism problem for a special class of digraphs.