A partially ordered set of functionals corresponding to graphs

Abstract For a graph G whose vertices are v1,v2,…,vm and where E is the set of edges, we define a functional U G (h)=ʃʃ…ʃ ∏ {v i ,v j }∈E h(x i ,x j ) dμ(x 1 )dμ(x 2 )…dμ(x m ) , where h is a nonnegative symmetric function of two variables. We consider a binary relation ≽ for graphs with fixed numbers of vertices and edges, where G≽L means that UG(h)⩾UL(h) for every h. We prove that this relation is equivalent to the condition: the number of homomorphisms into every graph H from G is not less than from L. We obtain comparability and incomparability criteria and investigate the poset of k-edge trees. In particular, the first and the second maximal elements of this poset are found.