Related papers: On graphs with 2 trivial distance ideals
Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae…
Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…
We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…
A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is…
Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$…
A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$…
In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized.…
The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many…
This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…
The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every…
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…
The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the…
The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the…
A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
A 2-factor of a graph is a 2-regular spanning subgraph. For a graph $G$ and an independent set $I$ of $G$, let $\delta_G(I)$ denote the minimum degree of vertices contained in $I$. We show that (1) if every independent set $I$ of $G$…
Let $D(G)=(d_{ij})_{n\times n}$ denote the distance matrix of a connected graph $G$ with order $n$, where $d_{ij}$ is equal to the distance between vertices $v_{i}$ and $v_{j}$ in $G$. A graph is called distance integral if all eigenvalues…