Sidorenko’s conjecture for subdivisions and theta substitutions

The famous Sidorenko’s conjecture asserts that for every bipartite graph H , the number of homomorphisms from H to a graph G with given edge density is minimised when G is pseudorandom. We prove that for any graph H , a graph obtained from replacing edges of H by generalised theta graphs consisting of even paths satisfies Sidorenko’s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko’s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].