Related papers: Radio numbers for generalized prism graphs

Let $G$ be a simple undirected graph. The regular number of $G$ is defined to be the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain…

Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than $1$, the greatest common divisor of the set of labels in…

An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ a function $f$ from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get…

A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident…

A proper labelling of a graph $G$ is a pair $({\pi},c_{\pi})$ in which ${\pi}$ is an assignment of numeric labels to some elements of $G$, and $c_{\pi}$ is a colouring induced by ${\pi}$ through some mathematical function over the set of…

Let $\mathbb{G}_{n,\gamma}$ be the set of simple and connected graphs on $n$ vertices and with domination number $\gamma$. The graph with minimum spectral radius among $\mathbb{G}_{n,\gamma}$ is called the minimizer graph. In this paper, we…

A total prime labeling of a graph of order $n$ is an extension of a prime labeling in which we distinctly label the vertices and edges. The goal of the labeling is for adjacent vertex labels to be relatively prime, and for each vertex of…

Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $\gamma(G)$, $\gamma_{\rm pr}(G)$, and $\gamma_{t}(G)$, respectively. For…

Given a graph, an $L(p,1)$-labeling of the graph is an assignment $f$ from the vertex set to the set of nonnegative integers such that for any pair of vertices $(u,v),|f (u) - f (v)| \ge p$ if $u$ and $v$ are adjacent, and $f(u) \neq f(v)$…

A $k$-L(2,1)-labeling of a graph is a function from its vertex set into the set $\{0,...,k\}$, such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. It is known…

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$…

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum…

A simple $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$ such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex…

A graph G is distinguished if its vertices are labelled by a map \phi: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves \phi. The distinguishing number of G is the minimum number k necessary for \phi to distinguish…

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ by…

We consider the multi-broadcast problem in arbitrary connected radio networks consisting of $n$ nodes. There are $k$ designated source nodes for some fixed $k \in \{1,\ldots,n\}$, and each source node has a distinct piece of information…

The number of nodes of a network, called its size, and the largest distance between nodes of a network, called its diameter, are among the most important network parameters. Knowing the size and/or diameter is a prerequisite of many…

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors…

Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and…

The radio $k$-chromatic number $rc_k(G)$ of a graph $G$ is the minimum integer $\lambda$ such that there exists a function $\phi: V(G) \to \{0,1,\cdots, \lambda\}$ satisfying $|\phi(u)-\phi(v)| \geq k+1 - d(u,v)$, where $d(u,v)$ denotes the…