Related papers: Quasi-({\lambda}; n)-distance-balanced graphs

We show that every cubic graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree deviates from $\frac{n}{4}$ by at most $\frac{1}{2}$, up to three exceptions. This resolves the conjecture of Alon…

In this paper we consider a distance-regular graph $\Gamma$. Fix a vertex $x$ of $\Gamma$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\mathbb{C}$-algebra generated by the Bose-Mesner algebra $M$ of…

Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems…

The \textit{biharmonic distance} (BD) is a fundamental metric that measures the distance of two nodes in a graph. It has found applications in network coherence, machine learning, and computational graphics, among others. In spite of BD's…

It is provided a local characterization of quasi-crystal graphs, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems. It is also shown that quasi-crystal graphs…

The distance $d(u,v)$ between the vertices $u$ and $v$ of a connected graph $G$ is defined as the number of edges in a minimal path connecting them. The \emph{transmission} of a vertex $v$ of $G$ is defined by $\sigma(v)=\sum\limits_{u\in…

Due to exponential growth of complex data, graph structure has become increasingly important to model various entities and their interactions, with many interesting applications including, bioinformatics, social network analysis, etc.…

Let D be the distance matrix of a connected graph G and let nn(G), np(G) be the number of strictly negative and positive eigenvalues of D respectively. It was remarked in [1] that it is not known whether there is a graph for which np(G) >…

We focus on strongly connected, strong for short, digraphs since in this setting distance is defined for every pair of vertices. Distance ideals generalize the spectrum and Smith normal form of several distance matrices associated with…

The notion of $f$-ideal is recent and has so far been studied in several papers. In \cite{qfi}, the idea of $f$-ideal is generalized to quasi $f$-ideals, which is much larger class than the class of $f$-ideals. In this paper, we introduce…

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices $u$ and $v$, the oracle returns the shortest path distance between $u$…

An $\epsilon$-distance-uniform graph is one in which from every vertex, all but an $\epsilon$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in…

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C^*$-algebra $\mathcal{O}_{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}_{G,\Lambda}$…

Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix…

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…

The following problem has been proposed in [Research problems from the Aveiro workshop on graph spectra, {\em Linear Algebra and its Applications}, {\bf 423} (2007) 172-181.]:\\ (Problem AWGS.4) Let $G_n$ and $G'_n$ be two nonisomorphic…

Let $(X,\mathbf{d})$ be a metric space, $V\subseteq X$ a finite set, and $E \subseteq V \times V$. We call the graph $G(E,V)$ a {\em metric} graph if each edge $(u,v) \in E$ has weight $d(u,v)$. In particular edge $(u,u)$ is in the graph…

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance…

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since…