Explicit Boij-Soderberg theory of ideals from a graph isomorphism reduction

TL;DR

This paper explores how Betti table decompositions in Boij-Soderberg theory can distinguish non-isomorphic graphs by linking algebraic invariants to classical graph statistics, building on graph isomorphism reductions.

Contribution

It provides an explicit Boij-Soderberg decomposition framework for edge ideals of graphs, connecting algebraic invariants to graph isomorphism problems.

Findings

Betti table decompositions relate to classical graph statistics

Explicit formulas for Boij-Soderberg coefficients are derived

Method distinguishes non-isomorphic graphs via algebraic invariants

Abstract

In the origins of complexity theory Booth and Lueker showed that the question of whether two graphs are isomorphic or not can be reduced to the special case of chordal graphs. To prove that, they defined a transformation from graphs G to chordal graphs BL(G). The projective resolutions of the associated edge ideals is manageable and we investigate to what extent their Betti tables also tell non-isomorphic graphs apart. It turns out that the coefficients describing the decompositions of Betti tables into pure diagrams in Boij-Soderberg theory are much more explicit than the Betti tables themselves, and they are expressed in terms of classical statistics of the graph G.