Is there an algebraic way to determine from the adjacency matrix $A$ of a simple graph $G$, whether $G$ contains an induced path of fixed length $n$? I am particularly interested in the case $n=6$.
If there is no characterization, I would also be interested in (possibly strict) sufficient conditions on $A$, which forces $G$ to contain an induced $P_n$ (resp. $P_6$).
Thank you in advance!