Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

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.