Graded Betti numbers of some bipartite graphs

For [Formula: see text], a [Formula: see text]-firecracker graph, denoted by [Formula: see text], is obtained by concatenation of [Formula: see text] [Formula: see text]-star graphs by connecting one leaf from each [Formula: see text]-star graph. Given a finite simple graph [Formula: see text], one can associate a simplicial complex [Formula: see text]. In this paper, we compute all the graded Betti numbers of the edge ideal [Formula: see text] of the firecracker graph [Formula: see text] by using the combinatorial data associated with the simplicial complex [Formula: see text] We also find the regularity of the ideals [Formula: see text]. Further, using the domination parameters of the graphs, we explicitly compute the projective dimension of [Formula: see text].