Related papers: Strong resolving graphs: the realization and the c…

Let $G=(V(G),E(G))$ be a simple graph. A set $S \subseteq V(G)$ is a strong edge geodetic set if there exists an assignment of exactly one shortest path between each pair of vertices from $S$, such that these shortest paths cover all the…

With the prevalence of social media, the connectedness between people has been greatly enhanced. Real-world relations between users on social media are often not limited to expressing positive ties such as friendship, trust, and agreement,…

We present a simple iterative strategy for measuring the connection strength between a pair of vertices in a graph. The method is attractive in that it has a linear complexity and can be easily parallelized. Based on an analysis of the…

We present a different way to obtain generators of metric spaces having the property that the ``position'' of every element of the space is uniquely determined by the distances from the elements of the generators. Specifically we introduce…

We investigate graphs that have characteristic-dependent well-covered dimension and show how more of these graphs can be constructed from known ones.

Let $G=(V,E)$ be a graph of order $n$. A closed distance magic labeling of $G$ is a bijection $\ell \colon V(G)\rightarrow \{1,\ldots ,n\}$ for which there exists a positive integer $k$ such that $\sum_{x\in N[v]}\ell (x)=k$ for all $v\in V…

We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,\varphi)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph…

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric…

Let $G$ be a graph with $n$ vertices and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove…

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…

The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the…

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained…

An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is…

Let $G$ be a connected graph and $u,v$ and $w$ vertices of $G$. Then $w$ is said to {\em strongly resolve} $u$ and $v$, if there is either a shortest $u$-$w$ path that contains $v$ or a shortest $v$-$w$ path that contains $u$. A set $W$ of…

A vertex $v\in V(G)$ is said to distinguish two vertices $x,y\in V(G)$ of a nontrivial connected graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subset V(G)$ is a local metric generator for…

Signed graphs are an emergent way of representing data in a variety of contexts where antagonistic interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability for…

The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric…

Mining graph data has become a popular research topic in computer science and has been widely studied in both academia and industry given the increasing amount of network data in the recent years. However, the huge amount of network data…

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a…

In this book authors for the first time introduce the notion of strong neutrosophic graphs. They are very different from the usual graphs and neutrosophic graphs. Using these new structures special subgraph topological spaces are defined.…