Binomial Edge Ideals of Generalized block graphs

TL;DR

This paper classifies generalized block graphs based on the uniqueness of their extremal Betti number and establishes bounds on the regularity of their binomial edge ideals, linking algebraic properties to graph structure.

Contribution

It provides a classification of generalized block graphs with unique extremal Betti numbers and derives new bounds on the regularity of their binomial edge ideals.

Findings

Classification of generalized block graphs with unique extremal Betti number.

Lower bound on regularity based on minimal cut sets.

Improved upper bound on regularity using maximal cliques and pendant vertices.

Abstract

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where $m(G)$ is the number of minimal cut sets of the graph $G$ and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of $G$.