Related papers: 2-uniform words: cycle graphs, and an algorithm to…

In this paper, we study the word-representability of well-partitioned chordal graphs using split decomposition. We show that every component of the minimal split decomposition of a well-partitioned chordal graph is a split graph. Thus we…

The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the…

Representing words by vectors, or embeddings, enables computational reasoning and is foundational to automating natural language tasks. For example, if word embeddings of similar words contain similar values, word similarity can be readily…

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…

Graph-based semantic representations are valuable in natural language processing, where it is often simple and effective to represent linguistic concepts as nodes, and relations as edges between them. Several attempts has been made to find…

We give a necessary condition for two diagrams of $3$-regular spatial graphs with the same underlying abstract graph $G$ to represent isotopic spatial graphs. The test works by reading off the writhes of the knot diagrams coming from a…

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor…

We introduce the notion of a subgraph generated by an $R$-word $r$ of the Sch\"{u}tzenberger graph of a positive word $w$, $S\Gamma(w)$, where $w$ contains $r$ as its subword. We show that the word problem for a finitely presented Adian…

Let $\core G$ and $\corona G$ denote the intersection and the union, respectively, of all maximum independent sets of a graph $G$. In this work, we show that for a graph with at most two odd cycles, $\a{\core G}+\a{\corona G}$ is equal to…

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition…

Many complex questions in biology, physics, and mathematics can be mapped to the graph isomorphism problem and the closely related graph automorphism problem. In particular, these problems appear in the context of network visualization,…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper we prove that if $G$ is a unicyclic graph then for all $s \geq 1$ the regularity of $I(G)^s$ is exactly $2s+\text{reg}(I(G))-2$. We also…

A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation…

A graph $G$ is called universal for a family of graphs $\mathcal{F}$ if it contains every element $F \in \mathcal{F}$ as a subgraph. Let $\mathcal{F}(n,2)$ be the family of all graphs with maximum degree $2$. Ferber, Kronenberg, and Luh…

Recently, Jones et al. introduced the study of $u$-representable graphs, where $u$ is a word over $\{1,2\}$ containing at least one 1. The notion of a $u$-representable graph is a far-reaching generalization of the notion of a…

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an…

For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq…

The family of graphs of reduced words of a certain subcollection of permutations in the union $\cup_{n\geq 4}\frak{S}_{n}$ of symmetic groups is investigated. The subcollection is characterised by the hook cycle type $(n-2,1,1)$ with…

Let G=(V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v in V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v in V, there is…

We show that determining if an $n$-vertex graph has twin-width at most 4 is NP-complete, and requires time $2^{\Omega(n/\log n)}$ unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that $n$-vertex…