Let \(k\geq 3\) be a positive integer and \(n_1, \dots , n_k\) be a sequence of positive integers. The generalized theta graph \(G=\theta _ {n_1,\dots,n_k}\) is the graph constructed by \(k\) paths with \(n_1, \dots , n_k\) vertices with only the endpoints in common so that at most one of \(n_1, \dots , n_k\) equals to two. In the paper under review firstly, by separating into seven cases in terms of the lengths \(n_1, \dots, n_k\), the authors could find some upper bound for \(\mathrm{ara}(I(G))\). Then by using some characterizations of \(\mathrm{bigheight}(I(G))\) obtained in [S. M. Seyyedi and F. Rahmati, “Regularity and projective dimension of the edge ideal of generalized theta graph” (to appear)] some upper bounds are given for \(\mathrm{ara}(I(G))-\mathrm{bigheight}(I(G))\).
After that \(\mathrm{height}(I(G))\) is computed for this class of graphs in three cases. This leads the authors to characterize Cohen-Macaulayness and unmixedness of these graphs by separating the proof into seven cases. As a consequence of this paper, one can get to a nice corollary which says that “a generalized theta graph \(G\) is Cohen-Macaulay (unmixed) if and only if \(G=\theta _{2,3,4}\)”.