Linear strand of edge ideals of zero divisor graphs of the ring ℤ_n

For a simple graph G with edge ideal I(G), we study the (Formula presented.) -graded Betti numbers in the linear strand of the minimal free resolution of (Formula presented.) where (Formula presented.) is the zero divisor graph of the ring (Formula presented.). We present sharp bounds for the Betti numbers of (Formula presented.) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of (Formula presented.) for (Formula presented.) and pqr, with primes (Formula presented.).