Related papers: Spectra of Graphs and Closed Distance Magic Labeli…

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 graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of…

A graph $G=(V,E)$ is said to be distance magic if there is a bijection $f$ from a vertex set of $G$ to the first $|V(G)|$ natural numbers such that for each vertex $v$, its weight given by $\sum_{u \in N(v)}f(u)$ is constant, where $N(v)$…

Let $\Gamma=(V,E)$ be a graph of order $n$. A {\em closed distance magic labeling} of $\Gamma$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x) = r$ for all…

For an arbitrary set of distances $D\subseteq \{0,1, \ldots, d\}$, a graph $G$ is said to be $D$-distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$,…

A $\Gamma$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$, for which there exists $\mu \in \Gamma$, such that the weight $w(x) =\sum_{y\in…

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same…

A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values…

For a set of distances $D$, a graph $G$ on $n$ vertices is said to be $D$-magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) = k$, where…

Given a graph $G$ with $n$ vertices and an Abelian group $A$ of order $n$, an $A$-distance antimagic labelling of $G$ is a bijection from $V(G)$ to $A$ such that the vertices of $G$ have pairwise distinct weights, where the weight of a…

A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave…

A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group $\gr$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every…

A graph of order $n$ is distance magic if it admits a bijective labeling of its vertices with integers from $1$ to $n$ such that each vertex has the same sum of the labels of its neighbors. In this paper we classify all distance magic…

For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where…

A graph is distance magic if it admits a bijective labeling of its vertices by integers from $1$ up to the order of the graph in such a way that the sum of the labels of all the neighbors of a vertex is independent of a given vertex. We…

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j}…

Hefetz, M\"{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $\Gamma$ be an Abelian group of order $n$. A…

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}\ell(y)$ of every vertex $x \in V$ is equal to the…

Let G=(V,E) be a graph of order n without isolated vertices. A bijection f:V -- {1,2,...n} is called a local distance antimagic labeling if the weights of any two adjacent vertices are not equal, where the weight of a vertex is defined to…

A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is…