Linear syzygy graph and linear resolution

For each squarefree monomial ideal $I\subset S=k[x_1,\dots ,x_n]$, we associate a simple graph $G_I$ by using the first linear syzygies of $I$. In cases, where $G_I$ is a cycle or a tree, we show the following are equivalent: (a) $I$ has a linear resolution (b) $I$ has linear quotients © $I$ is a variable-decomposable ideal In addition, with the same assumption on $G_I$, we characterize all monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension $2$ monomial ideals with a linear resolution. As an other application of our results, we also characterize all Cohen-Macaulay simplicail complexes in cases that $G_{\mathrm{\Delta }}\cong G_{I_{{\mathrm{\Delta }}^{\vee }}}$ is a cycle or a tree.