Distance Magic Graphs Research Articles

Distance magic labeling of join of graphs and big data analysis

Consider a simple graph G = 〈V, E〉 with n vertices. Define a function f from vertex set of G to set of integers 1, 2, …, n subject to the condition that there exists an integer k > 0 such that the sum of adjacent labels of each vertex in G is equal to k. In this paper, we prove that the graph K n - ⌊ n 2 ⌋ e has DML for all n ≥ 1, decomposition of distance magic graph K2n - {ne} into n ( n - 1 ) 2 edge distinct copies of C4 for all n ≥ 2 and DML of join of complete graphs gives unique mathematical model and it will be a useful model in big data analysis.

On Regular Distance Magic Graphs of Odd Order

Let G=(V,E) be a graph with n vertices. A bijection f:V→{1,2,…,n} is called a distance magic abeling f G if there exists an integer k such that ∑u∈N(v)f(u)=k for all v∈V, where N(v) is the set of all ertices adjacent to v. Any graph which admits a distance magic labeling is a distance magic graph. The existence of regular distance magic graphs of even order was solved completely in a paper by ronček, Kovář, and Kovářová. In two recent papers, the existence of 4-regular and of (n−3)-regular distance magic graphs of odd order was also settled completely. In this paper, we provide a similar classification of all feasible odd orders of r-regular distance magic graphs when r=6,8,10,12. Even though some nonexistence proofs for small orders are done by brute force enumeration, all the existence proofs are constructive.

Existence of Distance Magic Graph for Every Magic Constant

Existence of Distance Magic Graph for Every Magic Constant

Some distance magic graphs

A graph G=(V,E), where |V|=n and |E|=m is said to be a distance magic graph if there exists a bijection from the vertex set V to the set {1,2,…,n} such that, ∑v∈N(u)f(v)=k, for all u∈V, which is a constant and independent of u, where N(u) is the open neighborhood of the vertex u. The constant k is called the distance magic constant of the graph G and such a labeling f is called distance magic labeling of G. In this paper, we present new results on distance magic labeling of Cnr and neighborhood expansion Dn(G) of a graph G.

Zero Sum Partition of Abelian Groups into Sets of the Same Order and its Applications

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-zero-sum-partition property ($m$-ZSP-property) if $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots , A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, where $g_0$ is the identity element of $\Gamma$.In this paper we study the $m$-ZSP property of $\Gamma$. We show that $\Gamma$ has the $m$-ZSP property if and only if $m\geq 3$ and $|\Gamma|$ is odd or $\Gamma$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.

Constant sum partition of sets of integers and distance magic graphs

Constant sum partition of sets of integers and distance magic graphs

Some Distance Magic Graphs

Let G = (V, E) be a graph of order n. A bijective function f:V→{1,2,…,n} is said to be a distance magic labeling of G if for every v∈V, ∑x∈N(v)f(x)=k (a constant). A graph which admits such a labeling is said to be a distance magic graph. In this paper we study distance magic labeling for the neighborhood expansion Dp(G) of a graph G and present a method for embedding regular graphs into distance magic graphs.

Orientable Z_n-distance magic labeling of the Cartesian product of many cycles

The following generalization of distance magic graphs was introduced in [2]. A directed Z_n-distance magic labeling of an oriented graph $\overrightarrow{G}=(V,A)$ of order n is a bijection $\overrightarrow{\ell}\colon V \rightarrow Z_n$ with the property that there is a $\mu \in Z_n$ (called the magic constant) such that w(x)= \sum_{y\in N_{G}^{+}(x)} \overrightarrow{\ell}(y) - \sum_{y\in N_{G}^{-}(x)} \overrightarrow{\ell}(y)= \mu$ for every x \in V(G). If for a graph G there exists an orientation $\overrightarrow{G}$ such that there is a directed Z_n-distance magic labeling $\overrightarrow{\ell}$ for $\overrightarrow{G}$, we say that G is orientable Z_n-distance magic and the directed Z_n-distance magic labeling $\overrightarrow{\ell}$ we call an orientable Z_n-distance magic labeling. In this paper, we find orientable Z_n-distance magic labelings of the Cartesian product of cycles. In addition, we show that even-ordered hypercubes are orientable Z_n-distance magic.

On zero sum-partition of Abelian groups into three sets and group distance magic labeling

We say that a finite Abelian group Γ has the constant-sum-partition property into t sets (CSP ( t ) -property) if for every partition n = r 1 + r 2 + … + r t of n , with r i ≥ 2 for 2 ≤ i ≤ t , there is a partition of Γ into pairwise disjoint subsets A 1 , A 2 , …, A t , such that ∣ A i ∣ = r i and for some ν ∈ Γ , ∑ a ∈ A i a = ν for 1 ≤ i ≤ t . For ν = g 0 (where g 0 is the identity element of Γ ) we say that Γ has zero-sum-partition property into t sets (ZSP ( t ) -property). A Γ -distance magic labeling of a graph G = ( V , E ) with ∣ V ∣ = n is a bijection l from V to an Abelian group Γ of order n such that the weight w ( x ) = ∑ y ∈ N ( x ) l( y ) of every vertex x ∈ V is equal to the same element μ ∈ Γ , called the magic constant . A graph G is called a group distance magic graph if there exists a Γ -distance magic labeling for every Abelian group Γ of order ∣ V ( G )∣ . In this paper we study the CSP (3) -property of Γ , and apply the results to the study of group distance magic complete tripartite graphs.

Union of distance magic graphs

Union of distance magic graphs

Spectra of graphs and closed distance magic labelings

Let G=(V,E) be a graph of order n. A closed distance magic labeling of G is a bijection ℓ:V(G)→{1,…,n} for which there exists a positive integer k such that ∑x∈N[v]ℓ(x)=k for all v∈V, where N[v] is the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic labelings and the spectra of graphs. These results are then applied to the strong product of graphs with complete graph or cycle and to the circulant graphs. We end with a number theoretic problem whose solution results in another family of closed distance magic graphs somewhat related to the strong product.

Distance magic circulant graphs

Let G=(V,E) be a graph of order n. A distance magic labeling of G is a bijection ℓ:V→{1,2,…,n} for which there exists a positive integer k such that ∑x∈N(v)ℓ(x)=k for all v∈V, where N(v) is the neighborhood of v. In this paper we deal with circulant graphs Cn(1,p). The circulant graph Cn(1,p) is the graph on the vertex set V={x0,x1,…,xn−1} with edges (xi,xi+p) for i=0,…,n−1 where i+p is taken modulo n. We completely characterize distance magic graphs Cn(1,p) for p odd. We also give some sufficient conditions for p even. Moreover, we also consider a group distance magic labeling of Cn(1,p).

A Heuristic for Distance Magic Labeling

Abstract A distance magic labeling of a graph G with magic constant k is a bijection λ from the V(G) into {1, 2,. . ., |V(G)|}, such that ∑u∈N(v) λ(u) = k for every vertex v. Here we present a heuristic algorithm for finding distance magic graphs and utilise it to find all distance magic graphs with at most 9 vertices.

Distance Magic Labeling and Two Products of Graphs

Let $$G=(V,E)$$ be a graph of order $$n$$ . A distance magic labeling of $$G$$ is a bijection $$\ell :V\rightarrow \{1,\ldots ,n\}$$ for which there exists a positive integer $$k$$ such that $$\sum _{x\in N(v)}\ell (x)=k$$ for all $$v\in V $$ , where $$N(v)$$ is the neighborhood of $$v$$ . We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.

Note on Distance Magic Products $$G\circ C_4$$ G ∘ C 4

A distance magic labeling of a graph $$G=(V,E)$$ of order $$n$$ is a bijection $$l :V \rightarrow \{1, 2,\ldots , n\}$$ with the property that there is a positive integer $$k$$ (called magic constant) such that $$w(x) = k$$ for every $$x \in V$$ . If a graph $$G$$ admits a distance magic labeling, then we say that $$G$$ is a distance magic graph. In the case of non-regular graph $$G$$ , the problem of determining whether there is a distance magic labeling of the lexicographic product $$G\circ C_4$$ was posted in Arumugam et al. (J Indonesian Math Soc 11–26, 2011). We give necessary and sufficient conditions for the graphs $$K_{m,n}\circ C_4$$ to be distance magic. We also show that the product $$C^{(t)}_3\circ C_4$$ of the Dutch Windmill Graph and the cycle $$C_4$$ is not distance magic for any $$t>1$$ .

Note on group distance magic complete bipartite graphs

Abstract A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).

Handicap distance antimagic graphs and incomplete tournaments

Let G = (V, E) be a graph of order n. A bijection f: V → {1, 2,…, n} is called a distance magic labeling of G if there exists a positive integer μ such that for all v ϵ V, where N(v) is the open neighborhood of v. The constant μ is called the magic constant of the labeling f. Any graph which admits a distance magic labeling is called a distance magic graph. The bijection f: V → {1, 2,…, n} is called a d-distance antimagic labeling of G if for V = {v1, v2, …, vn} the sums form an arithmetic progression with difference d.We introduce a generalization of the well-known notion of magic rectangles called magic rectangle sets and use it to find a class of graphs with properties derived from the distance magic graphs. Then we use the graphs to construct a special kind of incomplete round robin tournaments, called handicap tournaments.

DISTANCE MAGIC GRAPHS - A SURVEY

Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V → {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>Σ f(u) = k </i> for all <i>v ε V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-26

Distance magic graphs of high regularity

A distance magic labeling of a graph G with n vertices is such a bijection f from the vertex set of G to the set of integers {1, 2, …, n} that for every vertex in G the sum of labels of all adjacent vertices gives the same value k. A graph that allows such a labeling is a distance magic graph.There is an elegant construction of r-regular distance magic graphs with an even number of vertices for all feasible values of r. For graphs of odd order certain necessary and certain sufficient conditions are known for the existence of a distance magic labeling. In this paper we show that an (n -3)-regular distance magic graph with n vertices exists iff n ≡ 3 (mod 6) and that its structure is determined uniquely.Moreover, we simplify the constructions from a recent paper by Fronček into a single construction and provide another sufficient condition for the existence a distance magic graph with an odd number of vertices.