相关论文: On the subdirect product of graph bundles

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…

This survey purports to be an elementary introduction to compactly presented groups, which are the analogue of finitely presented groups in the broader realm of locally compact groups. In particular, compact presentation is interpreted as a…

In this paper, we explore the concept of the ``matrix product of graphs," initially introduced by Prasad, Sudhakara, Sujatha, and M. Vinay. This operation involves the multiplication of adjacency matrices of two graphs with assigned labels,…

We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and…

Let $s$ be a positive integer. Our goal is to find all finite abelian groups $G$ that contain a $2$-subset $A$ for which the undirected Cayley graph $\Gamma(G,A)$ has diameter at most $s$. We provide a complete answer when $G$ is cyclic,…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent, and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. Let $ G $ be a finite group, and let $ S $ be a…

We consider a Cartesian product quantum graph $\Gamma_{n_1}\Box\Gamma_{n_2}$ with standard vertex conditions, and complete the decomposition of Hilbert space $L^2(\Gamma_{n_1}\Box\Gamma_{n_2})$ and the Laplacian $\mathscr{H}$ on it by…

In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be…

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop…

Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality…

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion…

We define a family of graphs we call dual systolic graphs. This definition comes from graphs that are duals of systolic simplicial complexes. Our main result is a sharp (up to constants) isoperimetric inequality for dual systolic graphs.…

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of…

One of the most interesting problems on maximum nullity (minimum rank) is to characterize $M(\mathcal{G})$ ($mr(\mathcal{G})$) for a graph $\mathcal{G}$. In this regard, many researchers have been trying to find an upper or lower bound for…

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than 1 to exactly one vertex of $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code…

Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation),…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

For each group G which decomposes into a finitary direct product of free groups of finite rank we construct a regular band B such that the free idempotent generated semigroup over B contains a maximal subgroup isomorphic to G. In…

We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and $\Gamma_C$ the Cayley graph of $G$ based on $C \cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded…