Related papers: On a Koolen -- Park inequality and Terwilliger gra…

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for…

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$ and $b\not=1$, $\alpha=b-1$. The condition on $\alpha$ implies that $\Gamma$ is formally self-dual. For $b=q^2$ we use the adjacency matrix and dual…

For a graph $\Gamma$ and group $G$, $G^\Gamma$ is the subgroup of $G^{|\Gamma|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $\Gamma$. There is a natural epimorphism $G^\Gamma \to…

We prove that for every exact discrete group $\Gamma$, there is an intermediate C*-algebra between the reduced group C*-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the…

We study cospectral vertices on finite graphs in relation to the echolocation problem on Riemannian manifolds. First, We prove a computationally simple criterion to determine whether two vertices are cospectral. Then, we use this criterion…

It was proved by Mineyev and Yaman that, if $(\Gamma, \Gamma')$ is a relatively hyperbolic pair, the comparison map $$ H_b^k(\Gamma, \Gamma'; V) \to H^k(\Gamma, \Gamma'; V) $$ is surjective for every $k \ge 2$, and any bounded…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes…

An "edge guard set" of a plane graph $G$ is a subset $\Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $\Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of…

Let $G$ be a graph and $\Gamma$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\Gamma$, denoted by $R(G, \Gamma)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to\Gamma$…

In this short paper, we establish relations between the domination number $\gamma$, the total domination number $\gamma_t$, and the connected domination number $\gamma_c$ of a graph. In particular, we prove upper and lower bounds for…

A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone…

A graph pair $(\Gamma, \Sigma)$ is called stable if $\aut(\Gamma)\times\aut(\Sigma)$ is isomorphic to $\aut(\Gamma\times\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. A graph is…

For a 1-tough graph $G$ we define $\sigma_3(G) = \min\{\deg(u) + \deg(v)+ \deg(w):$ $\{u, v, w\}$ is an independent set of vertices$\}$ and $NC2(G)=\min \{|N(u)\cup N(v)|: d(u,v)=2\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\geq…

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the…

The \emph{local edge-length ratio} of a planar straight-line drawing $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$ that share a common vertex. The \emph{global edge-length ratio} of $\Gamma$ is the…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends on the connectivity), and even better bounds hold for planar graphs with minimum…

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…