V k-Super Vertex Out-Magic Labeling of Digraphs

2018

Let G be a graph with p vertices and q edges. An Ek-super vertex magic labeling (Ek-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2, . . . , p + q} with the property that f(E(G)) = {1, 2, . . . , q} and for each v ∈ V (G), f(v) + wk(v) = M for some positive integer M . For an integer k ≥ 1 and for v ∈ V (G), let wk(v) = ∑ e∈Ek(v) f(e), where Ek(v) is the set of all edges which are at distance at most k from v. The graph G is said to be Ek-regular with regularity r if and only if |Ek(e)| = r for some integer r ≥ 1 and for all e ∈ E(G). A graph that admits an Ek-SVML is called Ek-super vertex magic (Ek-SVM). This paper contain several properties of Ek-SVML in graphs. A necessary and sufficient condition for the existence of Ek-SVML in graphs has been obtained. Also, the magic constant for Ek-regular graphs has been obtained. Further, we establish E2-SVML of some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs. MSC: 05C78.

Malaya Journal of Matematik

Let G be a simple graph with p vertices and q edges. A V-super vertex magic labeling is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} such that f (V (G)) = {1, 2,. .. , p} and for each vertex v ∈ V (G), f (v) + ∑ u∈N(v) f (uv) = M for some positive integer M. A V k-super vertex magic labeling (V k-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} with the property that f (V (G)) = {1, 2,. .. , p} and for each v ∈ V (G), f (v) + w k (v) = M for some positive integer M. A graph that admits a V k-SVML is called V k-super vertex magic. This paper contains several properties of V k-SVML in graphs. A necessary and sufficient condition for the existence of V k-SVML in graphs has been obtained. Also, the magic constant for E k-regular graphs has been obtained. Further, we study some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs which admit V 2-SVML.

2005

A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. Since its introduction by Sedláček in 1963, research in both magic and antimagic labelings has been growing fast. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling. In vertex labeling, we introduce super vertex magic total labeling and super (a, d)vertex antimagic total labeling, and present new results in these labelings. Additionally, we give vertex labeling schemes for some particular classes of graphs. Moreover, we introduce consecutive vertex magic total labeling and give new results concerning properties of consecutive vertex magic total labeling. In edge labeling, we present new results in super (a, d)-edge antimagic total labeling. We go on to introduce a consecutive edge magic total labeling and present some new vii viii results in this labeling. As in the vertex labeling chapter, we also give edge labeling schemes for some particular classes of graphs.

Malaya Journal of Matematik

Let G be a finite and simple (p, q) graph. An one-one onto function f : V (G) ∪ E(G) → {1, 2, 3,. .. , p + q} is called Vsuper vertex magic graceful labeling if f (V (G)) = {1, 2, 3,. .. , p} and for any vertex v ∈ V (G), ∑ u∈N(v) f (uv)− f (v) = M, where M is a whole number. For an integer k ≥ 1, let E k (v) = {e ∈ E(G) : the distance between e from v is less than or equal to k}. For v ∈ V (G), we define w k (v) = ∑ e∈E k (v) f (e). A V k-super vertex magic graceful labeling (V k-SVMGL) is a one-one function f from V (G) ∪ E(G) onto the set {1, 2, 3,. .. , p + q} such that f (V (G)) = {1, 2, 3,. .. , p} and for any element v ∈ V (G), we have w k (v) − f (v) = M, where M is a whole number. In this paper, we study several properties of V k-SVMGL and we identify an equivalent condition for the E k-regular graphs which admits V k-SVMGL. At last we identify some families of graphs which admit V 2-SVMGL.

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE, 2017

For any non-trivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) ! A − {0} such that, the vertex labeling f+ defined as f+(v) = Pf(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk, the group of integers modulo k and these graphs are referred as k-magic graphs. In this paper we prove that the graphs such as open star of shell, flower, double wheel, cylinder, wheel, generalised Petersen, lotus inside a circle and closed helm are Zk-magic graphs. Also we prove that super subdivision of any graph is Zk-magic.

Discrete Applied Mathematics, 2012

Let G be a finite simple graph with p vertices and q edges. A vertex magic total labeling is a bijection from V (G) ∪ E(G) to the consecutive integers 1, 2, 3,. .. , p + q with the property that for every u ∈ V (G), f (u) +  v∈N(u) f (uv) = k for some constant k. Such a labeling is E-super if f (E(G)) = {1, 2, 3,. .. , q}. A graph G is called E-super vertex magic if it admits a E-super vertex magic labeling. In this paper, we study some basic properties of such labelings and we establish E-super vertex magic labeling of some families of graphs. The main focus of this paper is on the E-super vertex magicness of H m,n and on some necessary conditions for a graph to be E-super vertex magic.

Control Systems and Computers, 2019

We have shown the connection between vertex labelings of magic graph and its overgraph. Also, we have introduced binary relation on the set of all D i-distance magic graphs, where D i ⊂ {0, 1, ..., d}, i = 1, 2, ... and proved, that it is equivalence relation. Therefore, we have explored the properties of the graphs, which are in this relation. Finally, we have offered the algorithm of constructing r-regular handicap graph G = (V, E) of order n, where n ≡ 0(mod8), r ≡ 1,3(mod4) and 3 ≤ r ≤ n-5.

An antimagic labeling of a graph G is a one-to-one correspondence between E(G) and {1, 2, . . . , |E|} such that for any two distinct vertices u and v, the sum of the labels assigned to edges incident to u is distinct from the sum of the labels assigned to edges incident to v. It was conjectured by Hartsfield and Ringel [9] in 1990 that every connected graph other than K 2 has an antimagic labeling. This conjecture attracted considerable attention.

2016

For any non-trivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) ! A − {0} such that, the vertex labeling f+ defined as f+(v) = Pf(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk, the group of integers modulo k and these graphs are referred as k-magic graphs. In this paper we prove that the graphs such as open star of shell, flower, double wheel, cylinder, wheel, generalised Petersen, lotus inside a circle and closed helm are Zk-magic graphs. Also we prove that super subdivision of any graph is Zk-magic.

2004

Let G be a finite simple graph with v vertices and e edges. A vertex-magic total labeling is a bijection λ from V (G)∪E(G) to the consecutive integers 1, 2, • • • , v +e with the property that for every x ∈ V (G), λ(x) + Σ y∈N (x) λ(xy) = k for some constant k. Such a labeling is super if λ(V (G)) = {1, • • • , v}. We study some of the basic properties of such labelings, find some families of graphs that admit super vertex-magic labelings and show that some other families of graphs do not.