Sidorenko's conjecture asserts that every finite bipartite graph 
with at least one graph edge and every nonempty finite simple graph 
satisfy
 |
where 
is the graph homomorphism density and
is the complete graph on two vertices.
Thus the bound counts arbitrary adjacency-preserving maps,
rather than only injective copies.
The inequality is established for trees, even cycle graphs, complete bipartite graphs,
and hypercube graphs (Conlon and Lee 2021). A
graph satisfying it for every 
is called a Sidorenko graph
(Zhao 2023). The 5-Möbius ladder, obtained
by deleting a Hamiltonian cycle from 
, is a classical test case (Zhao 2023).
