Spectral Sidorenko inequalities and edge-spectral supersaturation

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge \lambda(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $\lambda(G)>\lambda(S_{t-1,m})$, $$\texttt{#} K_{t,t}(G) \ge \Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{#}C_{2t}(G) \ge \Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities.