Related papers: Short incompressible graphs and $2$-free groups

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.

We establish combinatorial characterizations of virtually torsion-free and virtually free groups using the canonical graph decomposition theory in \cite{DJKK22}. Our main results show that a finitely presented, residually finite group…

In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to…

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all…

Whiteley \cite{wh} gives a complete characterization of the infinitesimal flexes of complete bipartite frameworks. Our work generalizes a specific infinitesimal flex to include joined graphs, a family of graphs that contain the complete…

Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space $\mathbb R^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition…

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic (in the Gromov sense)…

We study unknottedness for free boundary minimal surfaces in a three-dimensional Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary, and for self-shrinkers in the three-dimensional Euclidean space. For doing…

Let $L$ be a finite-dimensional Lie algebra over a field $F$. In This paper we introduce the \emph{nilpotent graph} $\Gamma_\mathfrak{N}(L)$ as the graph whose vertices are the elements of $L \setminus \nil(L)$, where \[\nil(L) = \{x \in L…

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic…

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…

The filling length of an edge-circuit \eta in the Cayley 2-complex of a finite presentation of a group is the least integer L such that there is a combinatorial null-homotopy of \eta down to a base point through loops of length at most L.…

In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is…

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a…

For a positive integer $d$, a connected graph $\Gamma$ is a symmetrical 2-extension of the $d$-dimensional grid $\Lambda^d$ if there exists a vertex-tran\-sitive group $G$ of automorphisms of $\Gamma$ and its imprimitivity system $\sigma$…

Let F be a finite field and R = M2(F) be 2x2 matrix ring over F. In this paper, we explicitly determine all the idempotents in R. Using these idempotents, we study the idempotent graph of R whose vertex set is the set of non-trivial…

We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges (2) has…

Let $G$ be a simply connected, connected completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}.$ Next, let $\pi$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a…

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of…

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this…