Hom Complexes Research Articles

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such ‘Hom complexes’ are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs G and H, we consider the polyhedral complex Hom⃗(G,H) that parametrizes the directed graph homomorphisms f:G→H. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed Hom⃗ complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom⃗ complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom⃗ complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom⃗(G,Tn) for Tn a tournament, obtaining a complete answer for the case of transitive Tn. If G=Tm is also a transitive tournament, we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom⃗ complexes.

Hom-Complex-Based Machine Learning (HCML) for the Prediction of Protein-Protein Binding Affinity Changes upon Mutation.

Protein-protein interactions (PPIs) are involved in almost all biological processes in the cell. Understanding protein-protein interactions holds the key for the understanding of biological functions, diseases and the development of therapeutics. Recently, artificial intelligence (AI) models have demonstrated great power in PPIs. However, a key issue for all AI-based PPI models is efficient molecular representations and featurization. Here, we propose Hom-complex-based PPI representation, and Hom-complex-based machine learning models for the prediction of PPI binding affinity changes upon mutation, for the first time. In our model, various Hom complexes Hom(G1, G) can be generated for the graph representation G of protein-protein complex by using different graphs G1, which reveal G1-related inner connections within the graph representation G of protein-protein complex. Further, for a specific graph G1, a series of nested Hom complexes are generated to give a multiscale characterization of the PPIs. Its persistent homology and persistent Euler characteristic are used as molecular descriptors and further combined with the machine learning model, in particular, gradient boosting tree (GBT). We systematically test our model on the two most-commonly used data sets, that is, SKEMPI and AB-Bind. It has been found that our model outperforms all the existing models as far as we know, which demonstrates the great potential of our model for the analysis of PPIs. Our model can be used for the analysis and design of efficient antibodies for SARS-CoV-2.

The lower extension groups and quotient categories

For certain full additive subcategories X of an additive category A, one defines the lower extension groups in relative homological algebra. We show that these groups are isomorphic to the suspended Hom groups in the Verdier quotient category of the bounded homotopy category of A by that of X. Alternatively, these groups are isomorphic to the negative cohomology groups of the Hom complexes in the dg quotient category A/X, where both A and X are viewed as dg categories concentrated in degree zero.

Warmth and edge spaces of graphs

In this paper we study a pair of numerical parameters associated to a graph G. On the one hand, we consider the (topological) connectivity of the polyhedral complex Hom(K2,G), a space of homomorphisms from a edge K2 into G. This approach dates back to the neighborhood complexes introduced by Lovász in his proof of the Kneser conjecture. In another direction we study the warmth ζ(G) of a graph G, a parameter introduced by Brightwell and Winkler based on the asymptotic behavior of d-branching walks in G and inspired by constructions in statistical physics. Both the warmth of G and the connectivity of Hom(K2,G) provide lower bounds on the chromatic number of G.Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph G is always less than three plus the connectivity of Hom(K2,G). We succeed in establishing a first nontrivial case of the conjecture, by showing that ζ(G)≤3 if Hom(K2,G) has an infinite first homology group. We also calculate warmth for a family of ‘twisted toroidal’ graphs that are important extremal examples in the context of Hom complexes. Finally we show that ζ(G)≤n−1 if a graph G does not have the complete bipartite graph Ka,b for a+b=n. This provides an analogue for a similar result in the context of Hom complexes.

Homotopy types of the Hom complexes of graphs

The Hom complex Hom(T,G) of graphs is a CW-complex associated to a pair of graphs T and G, considered in the graph coloring problem. It is known that certain homotopy invariants of Hom(T,G) give lower bounds for the chromatic number of G.For a fixed finite graph T, we show that there is no homotopy invariant of Hom(T,G) which gives an upper bound for the chromatic number of G. More precisely, for a non-bipartite graph G, we construct a graph H such that Hom(T,G) and Hom(T,H) are homotopy equivalent but χ(H) is much larger than χ(G). The equivariant homotopy type of Hom(T,G) is also considered.

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Given an acyclic twisting cochain π:C→A, from a curved dg coalgebra C to a dg algebra A, we show that the associated twisted hom complex Homkπ(C,A) has cohomology equal to the Hochschild cohomology of A, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra A, along with its cup product, is a subquotient of the tensor product algebra A!⊗A of A with its Koszul dual.

Morphism complexes of sets with relations

Let $r$ be a positive integer. An $r$-set is a pair $X= (V(X), R(X))$ consisting of a set $V(X)$ with a subset $R(X)$ of the direct product $V(X)^{r}$. The object of this paper is to investigate the Hom complexes of $r$-sets, which were introduced for graphs in the context of the graph coloring problem. In the first part, we introduce simplicial sets which we call singular complexes, and show that singular complexes and Hom complexes are naturally homotopy equivalent. The second part is devoted to the generalization of $\times$-homotopy theory established by Dochtermann. We show the folding theorem for hypergraphs which was partly proved by Iriye and Kishimoto.

Homology Theory of Graphs

Higher homotopy of graphs has been defined in several articles. In Dochterman (Hom complexes and homotopy theory in the category of graphs. arXiv math/0605275 v2,28/09/2006, 2006), the authors asked for a companion homology theory. We define such a theory for the category of unoriented reflexive graphs; it exhibits a long exact sequence for a pair of graphs (G, A), satisfies an excision property and a Hurewicz theorem. This allows us to compute the top homology of the graphical n-spheres showing that the theory is not trivial and is able to detect n-dimensional holes in a graph. The long-term objective is to compare the homotopy of the topological and graphical spheres.

Hom complexes and hypergraph colorings

Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.

Hom Complexes of Set Systems

A set system is a pair $\mathcal{S} = (V(\mathcal{S}),\Delta(\mathcal{S}))$, where $\Delta(\mathcal{S})$ is a family of subsets of the set $V(\mathcal{S})$. We refer to the members of $\Delta(\mathcal{S})$ as the stable sets of $\mathcal{S}$. A homomorphism between two set systems $\mathcal{S}$ and $\mathcal{T}$ is a map $f : V(\mathcal{S}) \rightarrow V(\mathcal{T})$ such that the preimage under $f$ of every stable set of $\mathcal{T}$ is a stable set of $\mathcal{S}$. Inspired by a recent generalization due to Engström of Lovász's Hom complex construction, the author associates a cell complex $\mathrm{Hom}(\mathcal{S},\mathcal{T})$ to any two finite set systems $\mathcal{S}$ and $\mathcal{T}$. The main goal of the paper is to examine basic topological and homological properties of this cell complex for various pairs of set systems.

Topology of Hom complexes and test graphs for bounding chromatic number

The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on the chromatic number. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, ⊙) and Hom(⊙, G) complexes as functors from graphs to posets, and introduce a functor (⊙)1 from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of Hom complexes in terms of spaces of equivariant poset maps and Γ-twisted products of spaces. When P:= F(X) is the face poset of a simplicial complex X, this provides a useful way to control the topology of Hom complexes. These constructions generalize those of the second author from [17] as well as the calculation of the homotopy groups of Hom complexes from [8].

SIMPLE ^|^Sigma;r-HOMOTOPY TYPES OF HOM COMPLEXES AND BOX COMPLEXES ASSIGNED TO r-GRAPHS

For a pair of graphs, Lovasz introduced a polytopal complex called the Hom complex in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Given an r-graph H, we compare the Hom complex of the complete r-partite r-graph and H with the box complex of H invented by Alon, Frankl and Lovasz. We verify that both complexes which are equipped with right actions of the symmetric group on r letters ∑_r, are of the same simple ∑_r -homotopy type.

The universality of Hom complexes of graphs

It is shown that given a connected graph T with at least one edge and an arbitrary finite simplicial complex X, there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials of graphs, and subdivisions are established that may be of independent interest.

Set Partition Complexes

The Hom complexes were introduced by Lovasz to study topological obstructions to graph colorings. The vertices of Hom(G,Kn) are the n-colorings of the graph G, and a graph coloring is a partition o...

Set Partition Complexes

The Hom complexes were introduced by Lovasz to study topological obstructions to graph colorings. The vertices of *Hom*(G,K n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that *Hom*(G,K n ) is (n−d−2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.

Hom complexes and homotopy in the category of graphs

Abstract In this extended abstract we develop a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product. We show that graph ×-homotopy is characterized by the topological properties of the so-called Hom complex, a functorial way to assign a poset to a pair of graphs. Along the way we establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘ A -theory’ associated to the cartesian product in the category of reflexive graphs. For proofs and further discussions we refer the reader to the full paper [Anton Dochtermann. Hom complexes and homotopy theory in the category of graphs. arXiv:math.CO/0605275 ].

Homotopy groups of Hom complexes of graphs

The notion of ×-homotopy from [Anton Dochtermann, Hom complexes and homotopy theory in the category of graphs, European J. Combin., in press] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom ∗ ( G , H ) with the homotopy groups of Hom ∗ ( G , H I ) . Here Hom ∗ ( G , H ) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and H I is the graph of based paths in H. As a corollary it is shown that π i ( Hom ∗ ( G , H ) ) ≅ [ G , Ω i H ] × , where ΩH is the graph of based closed paths in H and [ G , K ] × is the set of ×-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of [Eric Babson, Hélène Barcelo, Mark de Longueville, Reinhard Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (1) (2006) 31–44], where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of × -homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph × -homotopy is characterized by the topological properties of the Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; Hom complexes were introduced by Lovász and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph × -homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of × -homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘ A -theory’ associated to the cartesian product in the category of reflexive graphs.

A Physical Force may Expose Hox Genes to Express in a Morphogenetic Density Gradient

In both invertebrates and vertebrates, a set of homeobox genes is involved in the primary pattern formation along the anterior–posterior axis of the developing organism. In particular, the genes of the Hox/ HOM complex are located in a physical order in the 3′to 5′direction of the gene clusters. Furthermore, the vertebrate genes of the Hoxa and Hoxd clusters are expressed following the empirical rules of temporal and spatial collinearities: the genes are expressed one after the other according to their positional order and their domains of expression start anteriorly and move gradually towards more posterior locations along the developmental axis. The mechanism that controls this remarkable expression behaviour remains elusive. A proposed morphogen gradient model could justify the serial gene expression in space and time during vertebrate limb development. It is therefore likely that a morphogen concentration ordering might cause the sequential gene expression. I put forward this hypothesis and explore some possibilities that concentration-dependent physical forces might push the Hoxa,d clusters to an environment where the transcriptional activity of the genes is possible. The suggested mechanisms offer satisfactory concentration resolution for differential gene expression. Some experiments are proposed to test the presence of such forces. The verification of this hypothesis would provide a solution to the interpretation problem of the positional information theory in development. Furthermore, it would broaden our knowledge of how gene transcription can be triggered.

Setting limits on homeotic gene function: restraint of Sex combs reduced activity by teashirt and other homeotic genes.

Each of the homeotic genes of the HOM or HOX complexes is expressed in a limited domain along the anterior-posterior axis. Each homeotic protein directs the formation of characteristic structures, such as wings or ribs. In flies, when a heat shock-inducible homeotic gene is used to produce a homeotic protein in all cells of the embryo, only some cells respond by altering their fates. We have identified genes that limit where the homeotic gene Sex combs reduced (Scr) can affect cell fates in the Drosophila embryo. In the abdominal cuticle Scr is prevented from inducing prothoracic structures by the three bithorax complex (BX-C) homeotic genes. However, two of the BX-C homeotic genes, Ultrabithorax (Ubx) and abdominal-A (abd-A), have no effect on the ability of Scr to direct the formation of salivary glands. Instead, salivary gland induction by Scr is limited in the trunk by the homeotic gene teashirt (tsh) and in the last abdominal segment by the third BX-C gene, Abdominal-B (AbdB). Therefore, spatial restrictions on homeotic gene activity differ between tissues and result both from the regulation of homeotic gene transcription and from restraints on where homeotic proteins can function.