We prove the multiplicity bounds conjectured by Herzog-HunekeSrinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of forests, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We determine when equality holds in the conjectured bound, and verify that when equality holds, the resolution is pure. We characterize forests that have C...

‎let $x=left(‎ ‎begin{array}{llll}‎ ‎ x_1 & ldots & x_{n-1}& x_n‎ ‎ x_2& ldots & x_n & x_{n+1}‎ ‎end{array}right)$ be the hankel matrix of size $2times n$ and let $g$ be a closed graph on the vertex set $[n].$ we study the binomial ideal $i_gsubset k[x_1,ldots,x_{n+1}]$ which is generated by all the $2$-minors of $x$ which correspond to the edges of $g.$ we show that...

We determine the Castelnuovo-Mumford regularity of binomial edge ideals complement reducible graphs (cographs). For cographs with $n$ vertices maximum grows as $2n/3$. also bound by graph theoretic invariants and construct a family counterexamples to conjecture Hibi Matsuda.