Depth of Binomial Edge Ideals in terms of Diameter and Vertex Connectivity

TL;DR

This paper characterizes graphs where the lower and upper bounds for the depth of binomial edge ideals differ by one, and computes the exact depth for these graphs, advancing understanding of their algebraic properties.

Contribution

It provides a structural classification of graphs with a specific relation between bounds on the depth of binomial edge ideals and calculates their exact depth.

Findings

Graphs with $L(G)+1=U(G)$ are structurally classified.

Exact depth of $S/J_G$ is computed for these graphs.

The study deepens understanding of the algebraic invariants of binomial edge ideals.

Abstract

Let $G$ be a simple connected non-complete graph and $J_G$ be its binomial edge ideal in a polynomial ring $S$. Using certain invariants associated to graphs, say $U(G)$, Banerjee and N\'{u}\~{n}ez-Betancourt gave an upper bound for the depth of $S/J_G$, and Rouzbahani Malayeri, Saeedi Madani and Kiani obtained a lower bound, say $L(G)$. Hibi and Saeedi Madani gave a structural classification of graphs satisfying $L(G)=U(G)$. In this article, we give structural classification of graphs satisfying $L(G)+1=U(G)$. We also compute the depth of $S/J_G$ for all such graphs $G$.