Behaviour of the Normalized Depth Function

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ field. The $k$th power $I^{[k]}$ of $I$ is the ideal $S$ generated by all monomials belonging to $I^k$. biggest integer $k$ such that $I^{[k]}\ne(0)$ called grade and it denoted $\nu(I)$. $d_k$ minimum degree $I^{[k]}$. Then, $\text{depth}(S/I^{[k]})\ge d_k-1$ for $1\le k\le\nu(I)$. normalized depth function defined as $g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, It expected $g_I(k)$ non-increasing any $I$. In this article we study behaviour $g_{I}(k)$ under various operations on ideals. Our main result characterizes cochordal graphs $G$ edge $I(G)$ have $g_{I(G)}(1)=0$. They are precisely whose complementary graph $G^c$ connected has cut vertex. As far-reaching application, given integers s<m$ construct $\nu(I(G))=m$ $g_{I(G)}(k)=0$ if only $k=s+1,\dots,m$. Finally, show non-negative some ideal.