Super Edge-magic Graphs Research Articles

Results on super edge magic deficiency of somewell-known classes of finite graphs

A graph Ω(Λ, Γ) is considered super edge magic if there exists a bijective function ϕ:Λ(Ω)∪Γ(Ω)→ {1, 2, 3, …,|Λ(Ω)|+|Γ(Ω)|} such that ϕ(τ1)+(τ1τ2)+ ϕ(τ2) is a constant for every edge τ1τ2 ∈ Γ(Ω), and ϕ(Λ(Ω))= {1, 2, 3, …,|Λ(Ω)|}. Furthermore, the super edge magic deficiency of a graph Ω, denoted as μs (Ω), is either the minimum non-negative integer η such that Ω∪ηK1 is a super edge magic graph or +∞ if such an integer η does not exist. In this paper, we investigate the super edge magic deficiency of certain families of graphs.

On the (Consecutively) Super Edge-Magic Deficiency of Subdivision of Double Stars

LetGbe a finite, simple, and undirected graph with vertex setVGand edge setEG. A super edge-magic labeling ofGis a bijectionf:VG∪EG⟶1,2,…,VG+EGsuch thatfVG=1,2,…,VGandfu+fuv+fvis a constant for every edgeuv∈EG. The super edge-magic labelingfofGis called consecutively super edge-magic ifGis a bipartite graph with partite setsAandBsuch thatfA=1,2,…,AandfB=A+1,A+2,…,VG. A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency ofG, denoted byμsG, is either the minimum nonnegative integernsuch thatG∪nK1is super edge-magic or+∞if there exists no suchn. The consecutively super edge-magic deficiency of a graphGis defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.

Further results regarding super edge-magic graphs and solution to an open problem

Graph labeling admits a distinct place in Discrete Mathematics and Combinatorics. In 1998, Hikoe Enomoto et al. brought out the idea of super edge-magic (SEMC) labelling of graphs. In this article, we shall provide a complete solution to an open problem, regarding double stars St, t , proposed by W. D. Wallis & A. M. Marr [9], by exhibiting a super edge-magic labeling of St, t , for all possible values of t and t. We will also provide a SEMC labelling of union of cycles Cr being disjoint and various trees and exhibit super edge-magicness of some cases of Usmanian Ladders and planar graphs involving chains of C 5.

How to Construct New Reverse Super Edge-Magic Graphs from some old ones

How to Construct New Reverse Super Edge-Magic Graphs from some old ones

Edge Bimagic Labelings of Graphs K2, N, KN, 2, N, Bistar, nC4, and Double Wheel

Ngurahet. al. [2010] noted super magic coverings of fans and ladders. Ahmad et. al. [2011] enumerated super edge-magic labeling for the union of non-isomorphic graphs. Lopez et.al. [2012] suggested perfect super edge-magic graphs of path and cycle. Gao and Fan [2013] analyzed super edge-magic labelings of multi-star St(a1,…, an), Ali et. al. [2013] magic Labelings of type (a, b, c) of families of wheels.In this article, edge bimagiclabelings for bipartite complete graph, double bipartite complete graph, bistar merging with a path, one point union of finite copies of cycle having four vertices, and closed double wheel are found.

Construction of Reverse Super Edge Magic Total Graphs

In this paper, we investigate the adjacency matrix of reverse super edge magic vertex graph and use this graph to construct other reverse super edge magic graphs with the same edge weight set. Additionally, by combining known reverse super edge magic labelled graphs, we give a construction for a new reverse super edge magic graph

Super edge-magic labeling of graphs: deficiency and maximality

A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to {1, 2, 3, ..., p+q} such that f(x) + f(xy) + f(y) is a constant for every edge $xy \in E(G)$ and f(V(G)) = {1, 2, 3, ..., p}. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or +~ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.

STAR SUPER EDGE MAGIC DEFICIENCY OF GRAPHS

A graph G is called edge - magic if there is a bijec- tive function f : V (G)∪E(G) → {1, 2, . . . , |V (G)|+|E(G)|} such that for every edge xy ∈ E(G), f(x) + f(xy) + f(y) = c is a con- stant, called the valence of f. A graph G is said to be super edge - magic if f(V (G)) = {1, 2, . . . , |V (G)|}. Let G be a graph with p vertices with V (G) = {v1, v2, . . . , vp}. In G, every vertex vi is identified to the center vertex of Smi , for some mi ≥ 0, 1 ≤ i ≤ n, where S0 = K1 and the graph is denoted by G(m1,m2,...,mp). Let M(G) = {(m1,m2, . . . ,mp)|G(m1,m2,...,mp) is a super edge magic graph }. The star super edge magic deficiency Sμ∗(G) is defined as Sμ∗(G) = min(m1,,m2,...,mp)(m1 + m2 + · · · + mp) if M(G) 6= ∅; +∞ if M(G) = ∅. In this paper we determine the star super edge magic deficiency of certain classes of graphs.

A new labeling construction from the [formula omitted]-product

The ⊗h-product that is referred in the title was introduced in 2008 as a generalization of the Kronecker product of digraphs. Many relations among labelings have been obtained since then, always using as a second factor a family of super edge-magic graphs with equal order and size. In this paper, we introduce a new labeling construction by changing the role of the factors. Using this new construction the range of applications grows up considerably. In particular, we can increase the information about magic sums of cycles and crowns.

ON CONSTRUCTIONS OF NEW SUPER EDGE-MAGIC GRAPHS FROM SOME OLD ONES BY ATTACHING SOME PENDANTS

Baskoro et al. [1] provided some constructions of new super edge-magic graphs from some old ones by attaching 1, 2, or 3 pendant vertices and edges. In this paper, we introduce (q, m)-super edge-magic total labeling and we give a construction of new super edge-magic graphs by attaching n pendant vertices and edges under some conditions, for any positive integer n. Also, we give a constraint on our construction.

Cycle-based super edge-magic graphs and their super edge-magic sequences

In this paper, we have introduced a new concept of super edge-magic sequence (SEMS) of a super edge-magic graph (SEMG) with p vertices and q edges. The super edge-magic sequence of natural numbers is denoted by {xi}, 1 ≤ i ≤ q. This sequence need not to be monotonic. We also extend the notion of cycle iteratively for constructing new SEMS based on the 'sequence for cycle', from which we acquire some families like Petersen graph and its extension, web graph and its extension and others.

Special super edge-magic sequences on (q + 1,q) and (q,q) graphs

We introduced a new concept of super edge–magic sequence (SEMS) of a super edge-magic graph (SEMG) with p vertices and q edges. The super edge-magic sequence of natural numbers is denoted by < xi >,1 ≤ i ≤ q. This sequence need not to be monotonic. In this paper, we project SEMS in different direction i.e., in the application point of view. Primarily, we discuss two special types of super edge-magic sequences of a super edge-magic graph (p, q) on p = q + 1 and p = q also drive a particular family of graphs like tree and unicyclic graphs. We finish the paper, by presenting some additional interesting properties of super edge-magic sequences.

The Jumping Knight and Other (Super) Edge-Magic Constructions

Let G be a graph of order p and size q with loops allowed. A bijective function \({f:V(G)\cup E(G)\rightarrow \{i\}_{i=1}^{p+q}}\) is an edge-magic labeling of G if the sum \({f(u)+f(uv)+f(v)=k}\) is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that \({f(V(G))=\{i\}_{i=1}^{p}}\) then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which are related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product to get edge-magic labelings of some families of graphs.

NOVEL SIGN OF SUPER EDGE-MAGIC GRAPH

In this paper, we introduce a new concept of super edge-magic sequence (SEMS) of a super edge-magic graph (SEMG) with p vertices and q edges. The super edge-magic sequence of natural numbers is denoted by , 1 ≤ i ≤ q. This sequence need not to be monotonic. In this track, we also drive some families of super edge-magic graphs from fabrication of new super edge-magic sequences by considering additional parameter. We complete this paper by discussing the special case like monotonic sequences related to the super edge-magic sequence.

Enlarging the classes of super edge-magic 2-regular graphs

A graph G is called super edge-magic if there exists a bijective function f: V(G) ≼ E(G) → {1, 2,…,|V(G)| + |E(G)|} such that f(V(G)) = {1, 2,…,|V(G)|} and f (u) + f(v) + f(uv) is a constant for each uv ∊ E(G). A graph G with isolated vertices is called pseudo super edge-magic if there exists a bijective function f: V(G) → {1, 2,…, |V(G)|} such that the set {f (u) + f(v)|uv ∊ E(G)} ≼ {2f (u)|degG u = 0} consists of |E (G)| + |{u ∊ V(G) |degG u = 0}| consecutive integers. In this paper, we enlarge the classes of super edge-magic 2-regular graphs by presenting some constructions that allow us to generate large classes of super edge-magic 2-regular graphs from previously known super edge-magic 2-regular graphs or pseudo super edge-magic graphs.

Super Edge-Magic Models

In this paper, we generalize the concept of super edge-magic graph by introducing the new concept of super edge-magic models.

Super Edge Least-Magic Graphs

Abstract A (p, q)-graph G = (V, E) is said to be super edge—magic if there exists a bijection f fromV ∪ E to {1, 2, 3,…, p + q } with vertices maps to {1, 2, 3,…, p} such that for all edges uv of G, f(u) + f(v) + f(uv) is a constant and bijection so denned is called a super edge— magic labeling of G, For any super edge—magic labeling of G, there is a constant c(f) such that for all edges uv of G, f(u) + f(v) + f(uv) = c(f) and its range is p + q + 3 ≤ c(f) ≤ 3p. In this paper we study super edge—magic graphs with constant c(f) = p+q +3 for at least one f and such graphs are denned as super edge least—magic(SEL—magic) graphs. We investigate the following general results on the structure of SEL—magic graphs including a result, which determines all the regular SEL—magic graphs. (1) A SEL—magic graph is either a forest with exactly one nontrivial component, which is a star or has a triangle. (2) If an eulerian (p,q)-graph G = (V, E) is SEL—magic then q ≡ 0, 3(mod4). (3) The minimum vertex degree δ of any SEL—monograph is at most 3. (4) There are exactly three nontrivial regular graphs K2,K3 and K2 × K3 which are SEL—magic. Also we define level joined planar grid graph L J : P m × P n and prove that it is SEL—magic. Also we give a general method of constructing new SEL—magic graphs from any given SEL—magic graph.

The place of super edge-magic labelings among other classes of labelings

A ( p, q)-graph G is edge-magic if there exists a bijective function f: V( G)∪ E( G)→{1,2,…, p+ q} such that f( u)+ f( v)+ f( uv)= k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f( V( G))={1,2,…, p}. In this paper, we present some necessary conditions for a graph to be super edge-magic. By means of these, we study the super edge-magic properties of certain classes of graphs. We also exhibit the relationships between super edge-magic labelings and other well-studied classes of labelings. In particular, we prove that every super edge-magic ( p, q)-graph is harmonious and sequential (for a tree or q⩾ p) as well as it is cordial, and sometimes graceful. Finally, we provide a closed formula for the number of super edge-magic graphs.