Abstract
An n-crown $$\mathcal {C}_{n,n}$$ on 2n vertices is a graph obtained from complete bipartite graph $${K}_{n,n}$$ by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex $$\Delta (G)$$ . In this paper, we use combinatorial data from the associated simplicial complex $$\Delta (\mathcal {C}_{n,n})$$ of the crown graph $$\mathcal {C}_{n,n}$$ and give a formula to find Betti numbers of the form $$\beta _{i,i+1}$$ of edge ideals of $$\mathcal {C}_{n,n}$$ . We also present a formula to find a particular Betti of the edge ideal of a crown graph. We explicitly compute the projective dimension of the edge ideals of crown graphs using domination parameters of the graphs.
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