Related papers: Generalization of neighborhood complexes

Graph Neural Networks (GNNs) are almost universally built on a single primitive: the neighborhood. Regardless of architectural variations, message passing ultimately aggregates over neighborhoods, which intrinsically limits expressivity and…

The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S^4, and so contains all knot groups. We investigate when Campbell and…

We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n\trianglelefteq G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does not…

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of…

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a…

We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to…

We prove the non-existence of special generic maps on complex projective space as our extended new result. Simplest special generic maps are Morse functions with exactly two singular points on spheres, or Morse functions in Reeb's theorem,…

We introduce the class of outerspatial 2-complexes as the natural generalisation of the class of outerplanar graphs to three dimensions. Answering a question of O-joung Kwon, we prove that a locally 2-connected 2-complex is outerspatial if…

The Weisfeiler-Lehman graph kernels are among the most prevalent graph kernels due to their remarkable time complexity and predictive performance. Their key concept is based on an implicit comparison of neighborhood representing trees with…

In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this…

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…

Using a result of Vdovina, we may associate to each complete connected bipartite graph $\kappa$ a $2$-dimensional square complex, which we call a tile complex, whose link at each vertex is $\kappa$. We regard the tile complex in two…

In this short note we answer a query of Brodzki, Niblo, \v{S}pakula, Willett and Wright by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser proved that…

We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…

Let $r$ be an odd integer, and $k$ an even integer. In this note, we present $r$-regular graphs which have no $\{k,r-k\}$-factors for all $1\le k\le {r\over2}-1$. This gives a negative answer to a problem posed by Akbari and Kano recently.

Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v \in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex…

We give essentially tight bounds for, $\nu(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ of $k$ vertices in a graph with twin-width at most~$d$. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet…

Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. We provide a combinatorial proof of positivity for certain cluster variables in these algebras. This proof uses a hypergraph generalization of…

The K{\L}R conjecture of Kohayakawa, {\L}uczak, and R\"odl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G_{n,p}, for sufficiently large p : = p(n), satisfy an embedding lemma…

The theorem of Greenberg-Kazhdan-Drinfeld describes the formal neighborhood of a closed arc. After giving a complete proof with examples, two possible versions for the relative case of the theorem are discussed. Each one is shown to hold…