Reflection positivity, rank connectivity, and homomorphism of graphs

Edge-reflection positivity and weighted graph homomorphisms

B. Szegedy [Edge coloring models and reflection positivity, {\sl Journal of the American Mathematical Society} {\bf 20} (2007) 969--988] showed that the number of homomorphisms into a weighted graph is equal to the partition function of a complex edge-coloring model. Using some results in geometric invariant theory, we characterize for which weighted graphs the edge-coloring model can be taken to be real valued that is, we characterize for which weighted graphs the number of homomorphisms into them are edge-reflection positive. In particular, we determine explicitly for which simple graphs the…

Homomorphism Complexes and k-Cores

We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\displaystyle D(G)=\max_{H\subseteq G}δ(H)$. This is a strong generalization of a theorem of Cukić and Kozlov, in which $D(G)$ is replaced by the maximum degree $Δ(G)$. It also generalizes the graph theoretic bound for chromatic number, $\displaystyleχ(G)\leq D(G)+1$, as $\displaystyleχ(G)=\min\{ m:\text{Hom}(G,K_m)\neq\varnothing\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c/n$ for a…

Undecidability of linear inequalities in graph homomorphism densities

The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovász asks whether the opposite is also true. We answer this question and also a related question of Razbo…

On tension-continuous mapings

Tension-continuous (shortly TT) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. From another perspective, tension-continuous mappings are dual to the notion of flow-continuous mappings and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we define new constructions and operations for graphs (such as graphs Delta(G)) and give evidence for the complex relationship of homomorphisms and TT mappings. Particularly, solving an open problem, we display pairs of TT-comparable and homomorphi…

Warmth and connectivity of neighborhood complexes of graphs

In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $\text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lovász in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $ζ(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ an…

有向图的同态复形、重配置与同伦

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

本文系统建立有向图 Hom 复形理论，研究其拓扑性质、图运算、邻域复形、同调消没及同态重配置，并定义相关有向同伦。