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

Let G be a simple connected noncomplete graph and JG be its binomial edge ideal in a polynomial ring S. Using certain invariants associated to graphs, say U(G), Banerjee and Núñez-Betancourt gave an upper bound for the depth of S∕JG, 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∕JG for all such graphs G.