相关论文: The strong-connectivity of word-representable digr…

Let $(W,S)$ be a Coxeter system of type $A$, so that $W$ can be identified with the symmetric group $\mathrm{Sym}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions $\{\,(i,i+1)\mid 1\leqslant i\leqslant…

A graph $G$ is well-covered if all maximal independent sets are of the same cardinality. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. An…

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S.…

Several popular language models represent local contexts in an input text $x$ as bags of words. Such representations are naturally encoded by a sequence graph whose vertices are the distinct words occurring in $x$, with edges representing…

In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2000 and are the subject of a book published by Kitaev in 2015. Recently there has been…

The enhanced power graph of a group $G$ is a graph with vertex set $G,$ where two distinct vertices $x$ and $y$ are adjacent if and only if there exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ In this paper, we…

Graph-based text representation focuses on how text documents are represented as graphs for exploiting dependency information between tokens and documents within a corpus. Despite the increasing interest in graph representation learning,…

We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a $k$-vertex-connected orientation. We prove that a connectivity of order $O(k^2)$ suffices. As a key tool,…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

Let $G=(V(G),E(G))$ be a simple graph. A non-empty set $S\subseteq V (G)$ is a weakly connected dominating set in $G$, if the subgraph obtained from $G$ by removing all edges each joining any two vertices in $V (G)\setminus S$ is connected.…

Let $G$ be a connected graph. A vertex $w$ strongly resolves a pair $u$, $v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a strong resolving…

We investigate inflection structure of a synthetic language using Latin as an example. We construct a bipartite graph in which one group of vertices correspond to dictionary headwords and the other group to inflected forms encountered in a…

Given a connected graph $G$, a vertex $w\in V(G)$ distinguishes two different vertices $u,v$ of $G$ if the distances between $w$ and $u$ and between $w$ and $v$ are different. Moreover, $w$ strongly resolves the pair $u,v$ if there exists…

We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing…

Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this…

The presented work focuses on problems from determinant theory, set theory and topology. The term graph is the binding element that connects these problems. Graphs are distinguished by their geometrical simplicity, which helps in showing…

Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by…

The WL-rank of a digraph $\Gamma$ is defined to be the rank of the coherent configuration of $\Gamma$. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs…

We study a geometric representation problem, where we are given a set $\cal R$ of axis-aligned rectangles with fixed dimensions and a graph with vertex set $\cal R$. The task is to place the rectangles without overlap such that two…

A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a…