Labeling on Line Digraphs

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

In this paper it is shown that if a graph G possesses a spanning subgraph H with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then G also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order n, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order n connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

A one-to-one map X from E ∪ V onto the integers 1, 2, …, e + v is a vertex-magic total labeling if there is a constant h so that for every vertex x, λ(x) + ∑λ(xy) = h(3.1) where the sum is over all vertices y adjacent to x. So the magic requirement is wt(x) — h for all x. The constant h is called the magic constant for λ. Again, a graph with a vertex-magic total labeling will be called vertex-magic

The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings.

A conjecture on strong magic labelings of 2-regular graphs

Two new methods to obtain super vertex-magic total labelings of graphs

Consecutive magic graphs

Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.

Since Sedla$$\breve{\hbox {c}}$$cźek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study conse...

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.

In this study, we tried to test students’ deductive reasoning based on cognitive style in the application of super edge total local antimagic labels by applying Project Based Learning (PjBL) learning models. This study uses a mixed method, namely qualitative methods and quantitative methods. The research subjects consisted of two classes, namely 28 control class students and 30 experimental class students. The instruments used in this study were pretest, posttest, observation sheet, and interview sheet. Before the research activities, we tested the homogeneity of both classes using the results of the pretest. Homogeneity test results show the sig value is 0.481> 0.05, so the average difference between the two classes is not significant. This shows that both classes are homogeneous or have the same ability. Next, we also discuss the normality of using posttest data. This shows that the independent sample test on the posttest data from the control class and the experimental class obtained a sig score of 0.657 <0.05, so that it was significant which meant the difference between the control class and the experimental class. While the effect of punishment increases based on students’ cognitive style, it shows that 52% are in the very high category, 31% depend on the high category, 13% depend on the average category, and 4% depend on the low category. Project-Based Learning Model (PjBL) and the influence of deductive reasoning based on students’ cognitive styles in the super edge study of total local antimagic labeling.

A distance magic labeling of a graph G on p vertices is a bijection l from the vertex set of G to {1, 2, ···, p} such that for any vertex x of G, the weight of x, wG(x) = ∑v ∈ NG(x) l(v) is a constant. Further, if the weights of vertices of the graph G are in an arithmetic progression of the form a, a + d, …, a + (p – 1)d, then l is an (a, d)-distance antimagic labeling of the graph G. In this paper, we provide a partial solution to the problem on (a, d)-distance antimagicness of disconnected two regular graphs posted by Arumugam and Kamatchi [7].

Group distance magic and antimagic graphs

The application of Intuitionistic fuzzy total edge magic labelling to a graphical image at the 20th week of gestation provides insights into facilitates gender prediction as well as assessment of fetal blood flow on a second-trimester Doppler ultrasound screen. Ultrasounds screen the fetal heart rate during the 20th week of gestation using Doppler ultrasound for blood flow. A fetal heart rate above 2.5 beats per second suggests a female baby, while a rate less than 2.5 beats per second indicates a male baby. We convert the fetal heart blood flow into a graphical image and label it using intuitionistic fuzzy total edge magic labelling.

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 .