Edge-magic Total Labeling Research Articles

How to Construct Super Edge-Magic Total Labeling of Theta Graph θ(2,b,c)

This research study and provide the property of super edge-magic total labelings of theta graph. Edge magic labeling on a graph is an injective function from to a subset of integers { } with the property that there is a positive integer such as for each . An edge-magic labeling is called super edge-magic total labeling if it satisfies . A graph is called (super) edge-magic total if it admits some (super) edge-magic total labeling. A theta graph is constructed by embedding the endpoints of three paths of length consecutive such that there are two vertices of degree three and the other of degree two. This study gave some conditions for such a super edge-magic total of theta graph. Based on this condition, this paper introduce some algorithms to apply and develop super edge-magic total labeling from some previous theta graphs.

Super Edge-magic Total Strength of Some Unicyclic Graphs

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

Himpunan Kritis pada Graf Bintang

Labeling is a one-to-one mapping that maps each element of a graph to Positive numbers called labels. One of its kind is edge-magic total labeling. Under special conditions, the results set of labeled graphs whose subsets are labeled and positioned, which builds the same graph as the labeling, is called the critical set. To obtain the critical set of a graph we must know the type of graph. In this study is a star graph. This study aims to determine the critical set in star graphs. The star graph used is the K1.5 star graph using center points 1, n + 1 and 2n + 1. The research results show that by labeling the total magic side of the K1.5 star graph with center point λ(c) = 1, the magic number k=14 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = n+ 1, the magic number k=18 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = 2n + 1, the magic number k=22 is obtained. The possible critical set of K1.5 graphs is 120.

Strong vertex-magic and super edge-magic total labelings of the disjoint union of a cycle with 3-cycles

For more than half a century, the question of deciding which 2-regular graphs possess an edge-magic total labeling remains unsolved. A particularly difficult case involves those 2-regular graphs with many components isomorphic to C3, the cycle on three vertices. For 2-regular graphs, there is an obvious correspondence between an edge-magic total labeling and a vertex-magic total labeling (VMTL). The VMTL is strong or, it is a SVMTL if the smallest labels are used on the vertices. In this paper it is shown that each of the following disjoint unions has a SVMTL, and therefore a super edge-magic total labeling: for t≥4, C6∪(2t−1)C3, C8∪(2t−1)C3, C10∪(2t−1)C3 and 2C5∪(2t−1)C3; for t≥3, C9∪2tC3; for t≥2, C11∪2tC3. Combined with previous work, this now means that each disjoint union Cm∪sC3 of odd order possesses a SVMTL for each m such that 3≤m≤11, with the exception of three known graphs: C4∪C3, C5∪2C3 and C4∪3C3. This supports the conjecture stating that these are the only odd-order 2-regular graphs without a SVMTL. It also results in further evidence supporting MacDougall's conjecture regarding which regular graphs possess VMTLs, in general.It is a noteworthy feature of the proofs that a single Kotzig array is used to prove multiple theorems, providing a unifying element to labelings of different types of 2-regular graphs. The proofs are constructive, and this work is self-contained.

Labeling of 2-regular Graphs by Odd Edge Magic

An edge magic total labeling of a graph G(p, q) is said to be an Odd edge magic total labeling if λ(E) = {1,3, …2q – 1} with the condition that for each edge uv ∈ E, λ(u) + λ(uv) +λ(v) = ke, where ke is known as the magic constant. In this paper we resolute cycles of odd length, disjoint union of C3 ∪ C4r+2 (r ≥ 1), disjoint union of C4 ∪ C4r-1 (r > 1), disjoint union of C3 ∪ C4r (r > 1), disjoint union of C4 ∪ C4r-3 (r > 1), are Odd edge magic graphs.

On b-edge consecutive edge magic total labeling on trees

Let G = ( V , E ) be a simple, connected, and undirected graph, where V and E are the set of vertices and the set of edges of G . An edge magic total labeling on G is a bijection f : V ∪ E → {1, 2, …, | V |+| E |} , provided that for every u v ∈ E , w ( u v )= f ( u )+ f ( v )+ f ( u v )= K for a constant number K . Such a labeling is said to be a super edge magic total labeling if f ( V )={1,2,…,| V |} and a b -edge consecutive edge magic total labeling if f ( E )={ b +1, b +2,…, b +| E |} with b ≥ 1 . In this research, we give sufficient conditions for a graph G having a super edge magic total labeling to have a b -edge consecutive edge magic total labeling. We also give several classes of connected graphs which have both labelings.

On the construction of super edge-magic total graphs

Suppose G = ( V , E ) be a simple graph with p vertices and q edges. An edge-magic total labeling of G is a bijection f : V ∪ E → {1, 2, …, p + q } where there exists a constant r for every edge x y in G such that f ( x )+ f ( y )+ f ( x y )= r . An edge-magic total labeling f is called a super edge-magic total labeling if for every vertex v ∈ V ( G ) , f ( v )≤ p . The super edge-magic total graph is a graph which admits a super edge-magic total labeling. In this paper, we consider some families of super edge-magic total graph G . We construct several graphs from G by adding some vertices and edges such that the new graphs are also super edge-magic total graphs.

A Conjecture on Super Edge-Magic Total Labeling of 4-Cycle Books

A graph G is called cycle books B 4 , m , 2 if G consists of m cycles C 4 with a common path P 2 . Figueroa-Centeno, Ichishima, and Muntaner-Batle conjecture that the graph B 4 , m , 2 is super edge-magic total if and only if m is even or m ≡ 5 mod 8 . In this article, we prove this conjecture for m ≥ 36 and m = 0 mod (2).

A case study of an edge-magic total labeling of (a,b)-cycle books

Suppose α and β be the order and size of a graph G respectively. A one-one function h which maps the set of vertices and edges of a graph G onto the integers 1, 2, 3, …, α + β such that h(u) + h(uv) + h(v) = c for any edge (uv) ε E(G) is called an edge-magic total labeling of If h(u) ε {1,2, …, α} for any uεV(G) then h is a super edge-magic total labeling of G. One of interesting research topic is super edge-magic total labeling of cycle book. A cycle book B(a, m, b, n, t) is made up from m copies of cycle Ca and n copies cycle Cb with a commont path Pt . Super edge-magic total labeling of a cycle book B(a, m, b, n, 2) is still under investigation even for the case a = b This paper talk about a partial solution to this problem. We prove that a cycle book B(5, 2, 3, n, 2) has a super edge-magic total labeling for all positive integr n. In addition, we show that a cycle book B(5, 2, 3, n, 2) have an edge-magic total labeling for an integer n, 1 ≥ n ≥ 6.

Super magic deficiency of graphs

An edge-magic total (EMT) labeling for graph Γ is one-one map from π : V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|}, so that there manage a number c along a rule that for every edge, uv ∈ E(Γ), π (u) + π (uv) + π (v) = c. And if all vertices are assigned with positive integral numbers {1, 2, … ,|V(Γ)|} then this type of labeling is called a super EMT labeling. Super edge-magic, deficiency for graph Γ, represented as μs (Γ), which is least positive integral number n so that Γ∪ nK 1 has super EMT labeling, or ∞, if there does not exist n. In present article, compute a super edge-magic deficiency for the graph .

An application of super mean and magic graphs labeling on cryptography system

An edge-magic total (EMT) labeling on a simple graph G with p vertices and q edges is a bijection f: V∪E→{1, 2, …, p + q} such that for each edge xy of G, f(x) + f(xy) + f(y) = k, for a fixed positive integer k. Moreover, an EMT labeling f is a super edge-magic total labeling (SEMT) if f(V) = {1, 2, …, p}. Furthermore, a graph G is called a super mean (SM) graph if there exists an injective function f from the vertices of G to {0,1,2,…,q} such that when each edge uv is labeled with (f(u) +f(v))/2 when f(u) +f(v) is even, and (f(u) +f(v) + 1)/2 when f(u) +f(v) is odd, then the resulting edge labels are distinct. There are two essentials results that will be proposed in this paper. In the first result, we show that a graph (n, 1) − F Caterpillar has a SM labeling and a graph Hu,y has a SEMT labeling. In the second result, we will give an application of these labeling to increase the security level of Affine Cipher in which to encrypt a text on socials media.

SEMT valuation and strength of subdivided star of K 1,4

Abstract This study focuses on finding super edge-magic total (SEMT) labeling and deficiency of imbalanced fork and disjoint union of imbalanced fork with star, bistar and path; in addition, the SEMT strength for Imbalanced Fork is investigated.

On b-edge consecutive edge labeling of some regular tree

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em> E</em></span><span>) </span><span>be a finite (non-empty), simple, connected and undirected graph, where </span><span>V </span><span>and </span><span>E </span><span>are the sets of vertices and edges of </span><span>G</span><span>. An edge magic total labeling is a bijection </span><span>α </span><span>from </span><span><em>V</em> </span><span>∪ </span><span><em>E</em> </span><span>to the integers </span><span>1</span><span>, </span><span>2</span><span>, . . . , <em>n</em> </span><span>+ </span><em>e</em><span>, with the property that for every </span><span><em>xy</em> </span><span>∈ </span><em>E</em><span>, </span><span>α</span><span>(</span><em>x</em><span>) + </span><span>α</span><span>(</span><em>y</em><span>) + </span><span>α</span><span>(</span><em>xy</em><span>) = </span><em>k</em><span>, for some constant </span><em>k</em><span>. Such a labeling is called a </span><em>b</em><span>-edge consecutive edge magic total if </span><span>α</span><span>(</span><em>E</em><span>) = </span><span>{</span><span><em>b</em> </span><span>+ 1</span><span>, <em>b</em> </span><span>+ 2</span><span>, . . . , <em>b</em> </span><span>+ </span><em>e</em><span>}</span><span>. In this paper, we proved that several classes of regular trees, such as regular caterpillars, regular firecrackers, regular caterpillar-like trees, regular path-like trees, and regular banana trees, have a </span><em>b</em><span>-edge consecutive edge magic labeling for some </span><span>0 </span><span>< <em>b</em> < </span><span>|</span><span><em>V</em> </span><span>|</span><span>.</span></p></div></div></div>

RSEMT labeling of union of paths and subdivided star

A reverse edge magic total (REMT) labeling for the graph Γ = (V(Γ), E(Γ)) along with cardinality of p = |V(G)| and q = |E(G)| respectively. A one-one map π: V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|} having a rule that for every edge, uv ϵ E(Γ), then π (uv) − {π (u) + π (v)} = c, where c is a magic number. A reverse edge magic total (REMT) labeling is called a reverse super edge magic total (RSEMT) labeling if vertex-set receive natural numbers. In present paper, we compute the RSEMT labeling on the forest of paths and subdivided star.

On (a, d)-SEAT labeling of forests of subdivided stars

Graph ϒ = (V(ϒ), E(ϒ)) contain finite nodes V(ϒ) and finite edges E(ϒ). We also represent the order of the graph and size as μ = |V(ϒ)| and v = |V(ϒ)|. A graph is called (a, d)-edge magic total (EAT) labeling if there exists a bijective map ϕ from V(ϒ) ∪ E(ϒ) to the elements if the weight-set X = {ω(qr)|qr ∈ E(ϒ)} is arithmetic progression (A. P.) of positive integers which is started with a, d as common difference and ω(qr) = ϕ(q) + ϕ(qr) + ϕ(r). We say ω as the set of edge-weights. Further, if then the graph ϒ is said to be super (a, d)-edge antimagic total((a, d)-SEAT) labeling. In this article, we represent some new result on the (a, d)-SEAT labeling of some disjoint copies for subdivided star for the parameter .

On Edge Magic Total Labeling of (7, 3)-Cycle Books

A graph G is called (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] if G consists of m cycles Ca and n cycles Cb with a common path Pt. In this article we show that the (7, 3)-cycle books B[(C7, 1), (C3, n), P2] admits edge-magic total labeling. In addition we prove that (a, 3)-cycle books B[(Ca, 1), (C3, n), P2] admits super edge-magic total labeling for a = 7 and extends the values of a to 4x − 1 for any positive integer x. Moreover we prove that the (7, 3)-cycle books B[(C7, 2), (C3, n), P2] admits super edge-magic total labeling.

Edge magic total labeling of lexicographic product C4(2r+1) o ~K2 cycle with chords, unions of paths, and unions of cycles and paths

<p>An <em>edge magic total (EMT) labeling</em> of a graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is a bijection from the set of vertices and edges to a set of numbers defined by <span class="math"><em>λ</em> : <em>V</em> ∪ <em>E</em> → {1, 2, ..., ∣<em>V</em>∣ + ∣<em>E</em>∣}</span> with the property that for every <span class="math"><em>x</em><em>y</em> ∈ <em>E</em></span>, the weight of <span class="math"><em>x</em><em>y</em></span> equals to a constant <span class="math"><em>k</em></span>, that is, <span class="math"><em>λ</em>(<em>x</em>) + <em>λ</em>(<em>y</em>) + <em>λ</em>(<em>x</em><em>y</em>) = <em>k</em></span> for some integer <span class="math"><em>k</em></span>. In this paper given the construction of an EMT labeling for certain lexicographic product <span class="math">$C_{4(2r+1)}\circ \overline{K_2}$</span>, cycle with chords <span class="math"><em></em><sup>[<em>c</em>]<em>t</em></sup><em>C</em><sub><em>n</em></sub></span>, unions of paths <span class="math"><em>m</em><em>P</em><sub><em>n</em></sub></span>, and unions of cycles and paths <span class="math"> <em>m</em>(<em>C</em><sub><em>n</em><sub>1</sub>(2<em>r</em> + 1)</sub> ∪ (2<em>r</em> + 1)<em>P</em><sub><em>n</em><sub>2</sub></sub>)</span>.</p>

On edge magic total labeling of cycle book

The cycle book Bn is a graph with one copy of cycle C4 and n copies of cycle C3, and a common edge between C4 and n copies of C3. In this article we show that the cycle book Bn has an edge magic total labeling, Bn has an edge super magic total labelling for n = 1 and 2. Moreover we prove that B3 is not super edge magic total labeling.

Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar

AbstractA super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as $\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.

On Super Edge-magic Total Labeling of Modified Watermill Graph

An edge-magic total labeling on a graph G is one-to-one map from V(G) ∪ E(G) onto the set of integers 1,2, ...,ν + e, where ν = |V(G)| and e = |E(G)|, with the property that, given any edge uv, f(u) + f(u, ν}) + f(ν) = k for every u,v ∈ V(G), and k is called magic valuation. An edge-magic total labeling f is called super edge-magic total if f(v(G)) = {1,2 ...,|V(G)|} and f(E(G)) = {|V(G)| + 1, |V(G)| + 2,... |V(G) + E(G)|}. In this paper we investigate edge-magic total labeling of a new graph called modified Watermill graph. Furthermore, the magic valuation of the modified Watermill graph WM(n) is , for n odd, n ≥ 3.