Let \(R = \Bbbk [x_1,\ldots , x_n] \) be a polynomial ring over a filed \(\Bbbk \) . The edge ideal of a graph G is the monomial ideal, denoted by \(I_G\) , whose generators correspond to the edges in G. The quotient ring \(R/I_G\) is called the edge ring of G. In this paper, we study the minimal \(\mathbb {N}-\) graded free resolution of edge ring \(R/I_{\mathcal {C}_n^k}\) of a k-partite n-crown graph \(\mathcal {C}_n^k\) . We give a combinatorial formula for the graded Betti numbers in the linear strand of \(R/I_{\mathcal {C}_n^k}\) . We also determine the Hilbert series of the edge ring \(R/I_{\mathcal {C}_n^k}\) both in terms of i-faces of its independence complex and the graded Betti numbers of its edge ring. We then compare these Hilbert Series and derive a combinatorial formula for other graded Betti numbers of \(R/I_{\mathcal {C}_n^k}\) . As a consequence, we also determine the extremal graded Betti number of the edge ring of a \(k-\) partite \(n-\) crown graph \(R/I_{\mathcal {C}_n^k}\) . Moreover, we determine the projective dimension and the Castelnuovo–Mumford regularity of the edge rings of these graphs.
