Depth and Stanley depth of the path ideal associated to an $n$-cyclic graph

We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\equiv ̸1(mod4)$, tight bounds when $n\equiv 1(mod4)$ and for Stanley depth when $n\equiv 0,3,(\mbox{mod}\ 4)$,tightboundswhen$n\equiv 1,2,(\mbox{mod}\ 4)$. Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.