Cartesian Prime Graphs and Cospectral Families

We introduce a method for constructing larger families of connected cospectral graphs from two given cospectral families of sizes $p$ and $q$. The resulting family size depends on the Cartesian primality of the input graphs and can be one of $pq$, $p+q-1$, or $\max (p,q)$, based on the strictness of the applied conditions. Under the strictest condition, our method generates $O(p^3{q}^3)$ new cospectral triplets, while the more relaxed conditions yield $\Omega (p{q}^3+q{p}^3)$ such triplets. We also use the existence of specific cospectral families to establish that of larger ones.