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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.
Singh et al. (Mon,) studied this question.