Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs

TL;DR

This paper investigates the associative spectra of graph algebras, classifying undirected graphs into three types based on their spectra, and explores spectra for various directed graphs using homomorphisms of DFS trees.

Contribution

It introduces a classification of undirected graphs by their associative spectra and analyzes spectra for specific directed graph families, advancing the understanding of graph algebra properties.

Findings

Undirected graphs have three spectral types: 1, powers of 2, and Catalan numbers.

Associative and antiassociative digraphs are characterized.

Spectra for paths, cycles, and two-vertex graphs are determined.

Abstract

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.