Binomial edge ideals of crown graphs

Abstract: In this article, we explore the class of graphs for which the projective dimension of the quotient of the binomial edge ideals matches the big height of that ideal. Additionally, we investigate the Vasconcelos number of binomial edge ideals for cycles and crown graphs. We also provide proof for [Conjecture 4.13, 3], which is related to the Vasconcelos number of binomial edge ideals for cycles.