Related papers: Path lifting properties and embedding between RAAG…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…

Many important results in extremal graph theory can be roughly summarised as "if a triangle-free graph $G$ has certain properties, then it has a homomorphism to a triangle-free graph $\Gamma$ of bounded size". For example, bounds on…

We bound the volume of thick embeddings of finite graphs into the Heisenberg group, as well as the volume of coarse wirings of finite graphs into groups with polynomial growth. This work follows the work of Kolmogorov-Brazdin, Gromov-Guth…

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose…

The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported in the end vertices of the path which are too…

We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the linear…

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for…

The maximum cardinality of an induced $2$-regular subgraph of a graph $G$ is denoted by $c_{\rm ind}(G)$. We prove that if $G$ is an $r$-regular graph of order $n$, then $c_{\rm ind}(G) \geq \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$ and we…

Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and…

Let $ \Phi=(G, \varphi) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ \Delta $. The associated adjacency matrix and degree matrix are denoted by…

A regular bipartite graph $\Gamma$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\Gamma)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\Gamma)$ that stabilizes the…

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a…

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete…

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local…

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that…

We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $\pi_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive…

Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\in\{1,2,...,n-1\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$…

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…

A finite simplicial graph \Gamma determines a right-angled Artin group G_\Gamma, with generators corresponding to the vertices of \Gamma, and with a relation vw=wv for each pair of adjacent vertices. We compute the lower central series…

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph…