Maximizing subgraph counts in regular graphs

Given a graph $H$, we investigate the $d$-regular graphs $G$ with the highest $H$-density. We reframe the problem as a continuous optimization problem on the eigenvalues of $G$ by relating injective homomorphism numbers from $H$ and homomorphism numbers from quotient graphs of $H$. For almost all $H$, this relation has non-spectral terms, which require bounding by spectral terms in a way that is sharp at the optimal graph. For bipartite $H$ and $d$ large enough, we show $G$ consists of disjoint copies of $K_{d,d}$. For non-bipartite $H$ and $d$ sufficiently large, $G$ is a collection of disjoint $K_{d+1}$ graphs. For $H=C_5$ and $d=3$, disjoint Petersen graphs emerge.