In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a cactus graph. We also identify certain subclass attaining the upper bound.
Let G be a finite simple graph on n vertices and J G denote the corresponding binomial edge ideal in the polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ]. In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete k-partite graph, fan graph, multi-fan graph and wheel graph.
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