Certain homological invariants of some bipartite graphs

For integers [Formula: see text], the graph obtained by joining one leaf of each of [Formula: see text]-copies of [Formula: see text]-star graph with a single vertex distinct from all the vertices of [Formula: see text]-copies of [Formula: see text]-star graph is known as Banana tree graph, written as [Formula: see text]. In this paper, we shall investigate certain homological invariants of edge rings of Banana tree graphs [Formula: see text] in terms of associated combinatorial data. More precisely, we shall show that the regularity and projective dimension of edge ring of [Formula: see text] are [Formula: see text] and [Formula: see text], respectively, and deduce combinatorial formulae for computing all the nonzero graded Betti numbers of edge ring of [Formula: see text].