Related papers: D-Antimagic Labelings on Oriented Linear Forests

For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance multiset}, $D(S)$, is the multiset of pairwise distances between vertices of $S$ in $G$. Two vertex sets are called {\em homometric} if their distance multisets are…

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if…

An edge labeling of a connected graph $G = (V,E)$ is said to be local antimagic if it is a bijection $f : E \to \{1, . . . , |E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \ne f^+(y)$, where the induced vertex label…

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \to \{1,2,\ldots,q\}$ is called a local antimagic…

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that…

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…

The diameter of an undirected unweighted graph $G=(V,E)$ is the maximum value of the distance from any vertex $u$ to another vertex $v$ for $u,v \in V$ where distance i.e. $d(u,v)$ is the length of the shortest path from $u$ to $v$ in $G$.…

Given a graph $G=(V,E)$ and a colouring $f:E\mapsto \mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called…

Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) =…

An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges…

Let $G=(V,E)$ be a simple graph of size $m$ and $L$ a set of $m$ distinct real numbers. An $L$-labeling of $G$ is a bijection $\phi: E \rightarrow L$. We say that $\phi$ is an antimagic $L$-labeling if the induced vertex sum $\phi_+: V…

Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in…

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $ I_{\subseteq C} $, where $ C $ is a maximal strong vertex cover of $D$ and $ I_{\subseteq C} $ is the…

An arc-weighted digraph is a pair $(D,\omega)$ where $D$ is a digraph and $\omega$ is an \emph{arc-weight function} that assigns\ to each arc $uv$ of $D$ a nonzero real number $\omega(uv)$. Given an arc-weighted digraph $(D,\omega)$ with…

Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length…

Let T=(V,E) be a tree graph with non-negative weights defined on the vertices. A vertex z is called a separating vertex for u and v if the distances of z to u and v are not equal. A set of vertices L\subseteq V is a feasible solution for…

An antimagic {labeling} of a graph $G=(V,E)$ is a one-to-one mapping $f: E\rightarrow\{1,2,\ldots,|E|\}$, ensuring that the vertex sums in $V$ are pairwise distinct, where a vertex sum of a vertex $v$ is defined as the sum of the labels of…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

We show that for every $\eta>0$ every sufficiently large $n$-vertex oriented graph D of minimum semidegree exceeding $(1 + \eta) k/2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k \ge \eta n$. In…

Given a finite directed graph with $n$ vertices, we define a metric $d_G$ on $\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. The weight of a word is defined as the number of vertices that can be reached by a…