Abstract: A graph $X$ is said to be determined by its generalized spectrum if for any graph $H$, $H$ and $X$ being cospectral with cospectral complements implies that $H$ is isomorphic to $X$. This paper determines formulas for enumerating the number of paths with length no more than $5$ in any graph and proves that if $k\not\equiv2$ (mod 3), then the grid graph $P_k\square P_2$ is determined by its generalized spectrum.

Key words: generalized spectrum, generalized cospectral, determined by generalized spectrum, grid graph