VERTEX MAGIC AND EDGE MAGIC LABELINGS OF GRAPHS WITH GROUP ELEMENTS

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.

A reverse magic labelling of a graph G(V, E) is a bijection f: V ∪ E → {1, 2, 3, ......, v + ε} such that for all edges xy, f(xy) - {f(x) + f(y)} is a constant which is denoted by c(f). A reverse magic labelling of a graph G(V, E) is called reverse super edge-magic labelling of G if f(V) = {1, 2, ...... v} and f(E) = {v + 1, v + 2, ......, v + ε}. The reverse super edge-magic strength of a graph G,rsm(G), is defined as the minimum of all c(f) where the minimum is taken over all reverse edge-magic labelling f of G. In this paper we invented the reverse super edge-magic strength of banana trees.

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.

Distance magic graphs [formula omitted

Group distance magic and antimagic graphs

A $Γ$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $Γ$ of order $n$, for which there exists $μ\in Γ$, such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to $μ$. In this case, the element $μ$ is called the \emph{magic constant of} $G$. A graph $G$ is called a \emph{group distance magic} if there exists a $Γ$-distance magic labeling of $G$ for every Abelian group $Γ$ of order $n$. In this paper, we focused on cubic $Γ$-distance magic graphs as well as some properties of such graphs.

For any additive abelian group $A$. Let $\mu$ be an element of $A$, a graph $G=(V,E)$ is said to be $A$-vertex magic graph if there exist a labeling function $f:V(G)\rightarrow A\setminus\{0\}$ such that $\omega(v)=\sum_{u\in N(v)} f(u)=\mu$ for any vertex $v$ of $G$, where $N(v)$ is the set of the open neighborhood of $v$. In this paper, we prove that the graphs such as wheel, Corona $C_{n}\odot mk$, subdivision of ladder and $t$-fold wheel for $t\neq n$ nor $n-2$ are $A$-vertex magic graphs. Also we prove that the subdivide wheel, helm and closed helm are $Z_{k}$-vertex magic graphs. However we prove that the triangular book and $t$-fold wheel for $t=n,n-2$ are group vertex magic graphs.

In this paper, we consider A -vertex magic graphs, where A is a non-trivial Abelian group. We characterize Z -vertex magic graphs. We also explore the relation between the A -vertex magicness of a graph G and its reduced graph. In addition, we introduce a new type of labeling called A ′ -vertex magic labeling of graphs and characterize A -vertex magicness using A ′ -vertex magicness. We give a new procedure to embed any graph as an induced subgraph of an A -vertex magic graph and we construct infinite families of A -vertex magic graphs both of these procedure uses less number of vertices compared to the one given in (Sabeel et al. in Australas. J. Combin. 85(1) (2023), 49-60) and we also generalize some results proved in this paper. Finally we completely classify generalised friendship graph using A -vertex magicness and group vertex magicness.

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.