Density of linearity index in the interval of matching numbers

Given integers $2\le p\le c\le q$, we construct a finite simple graph $G$ with ${\nu }_1(G)=p$ and $\nu (G)=q$ for which the squarefree power $I(G{)}^{[k]}$ of the edge ideal $I(G)$ of $G$ has linear quotients for each $c\le k\le q$ and is not linearly related for each $1\le k<c$, where ${\nu }_1(G)$ is the induced matching number of $G$ and $\nu (G)$ is the matching number of $G$.