Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes

The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$\mathrm{.}Weprovethattheinequality$d \leq $\mathrm{r}\mathrm{e}\mathrm{g}$(\Delta) \cdot $\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}$(\Delta)$holdsforany$(d-1)$-dimensional Cohen-Macaulay simplicial complex $\mathrm{\Delta }$ satisfying $\mathrm{\Delta }=\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(\mathrm{\Delta })$, where $\mathrm{r}\mathrm{e}\mathrm{g}(\mathrm{\Delta })$ (resp. $\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}(\mathrm{\Delta })$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $\mathbb{k}[\mathrm{\Delta }]$. Moreover, for any given integers $d,r,t$ satisfying $r,t\ge 2$ and $r\le d\le rt$, we construct a Cohen-Macaulay simplicial complex $\mathrm{\Delta }(G)$ as an independent complex of a graph $G$ such that $\dim (\mathrm{\Delta }(G))=d-1$, $\mathrm{r}\mathrm{e}\mathrm{g}(\mathrm{\Delta }(G))=r$ and $\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}(\mathrm{\Delta }(G))=t$.