Some Cohen-Macaulay and unmixed binomial edge ideals

TL;DR

This paper investigates the algebraic properties of binomial edge ideals in specific graph classes, focusing on Cohen-Macaulayness and unmixedness, and extends existing results to generalized block graphs and certain graph products.

Contribution

It characterizes Cohen-Macaulay and unmixed binomial edge ideals for generalized block graphs and explores these properties in graph joins and corona products, generalizing prior work.

Findings

Computed the depth of binomial edge ideals for generalized block graphs

Characterized when these ideals are Cohen-Macaulay and unmixed for generalized block graphs

Analyzed unmixedness and Cohen-Macaulayness in graph joins and corona products

Abstract

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen-Macaulay and unmixed. So that we generalize the results of Ene, Herzog and Hibi on block graphs. Moreover, we study unmixedness and Cohen-Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs'.