We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly mappings). Existence of a mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see [10]). In this paper we study the relationship of the homomorphism and orders. W...

Motivated by the Alday-Gaiotto-Tachikawa (AGT) conjecture, we construct a homomorphism from affine super Yangian [Formula: see text] to universal enveloping algebra of rectangular W-superalgebra for all m ≠ n, m, n ≥ 2 or 3, = 0. Furthermore, show that image this is dense, provided k + ( − n)( l 1)

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special cl...

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is ‘sub-logarithmic’, then the range of such a homomorphism is super-constant. Furthermore, some exam...

Let G be an abelian topological group. The symbol Ĝ denotes the group of all continuous characters χ : G → T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that χ(E) ⊆ φ([−1/4, 1/4]) holds only for the trivial character χ ∈ Ĝ, where φ : R → T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most ...

Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between homomorphism densities to a linear inequality. Inspired by the work of Freedman,...