Group vertex magic labeling of some special graphs

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 $K_{m, n}$.

Klein cordial trees and odd cyclic cordial friendship graphs

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic} if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$ the group of integers modulo $k$. These $Z_k$-magic graphs are referred to as $k$-textit{magic} graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed helm graph, lotus inside a circle graph, $Godotoverline{K_m}$, $m$-splitting graph of a path and $m$-shadow graph of a path are $Z_k$-magic graphs.

Let G be a simple graph, let f : V(G)&#8594{1,2,...,|V(G)|} be a bijective mapping. The weight of v &#8712 V(G) is the sum of labels of all vertices adjacent to v . We say that f is a distance magic labeling of G if the weight of every vertex is the same constant k and we say that f is a handicap magic labeling of G if the weight of every vertex v is l + f(v) for some constant l. Graphs that allow such labelings are called distance magic or handicap, respectively. Distance magic and handicap labelings of regular graphs are used for scheduling incomplete tournaments. While distance magic labelings correspond to so called equalized tournaments, handicap labelings can be used to schedule incomplete tournaments that are more challenging to stronger teams or players, hence they increase competition and yield attractive schemes in which every games counts. We summarize known results on distance magic and handicap labelings and construct a new infinite class of 4-regular handicap graphs.

Let $V_4=\{0, a, b, c\}$ be the Klein-4-group with identity element 0 .A graph $G(V(G), E(G))$ is said to be neighbourhood $V_4$-magic if there exists a labeling $f: V(G) \rightarrow V_4 \backslash\{0\}$ such that the induced mapping $N_f^{+}: V(G) \rightarrow V_4$ defined by $N_f^{+}(v)=\sum_{u \in N(v)} f(u)$ is a constant map. If this constant is $p(p \neq 0)$, we say that $f$ is a $p$-neighbourhood $V_4$-magic labeling of $G$ and $G$ a $p$-neighbourhood $V_4$-magic graph. If this constant is zero, we say that $f$ is a 0 -neighghbourhood $V_4$-magic labeling of $G$ and $G$ a 0 -neighbourhood $V_4$-magic graph. In this paper we investigate middle graph of some special graphs that are $a$-neighbourhood $V_4$-magic, 0 -neighbourhood $V_4$-magic and both $a$-neighbourhood and 0 -neighbourhood $V_4$-magic.

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

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

Given a graph G with n vertices and m edges and an abelian group A of order equal to max(n,m). By labeling of A∗-vertex magic labeling and A∗-edge magic labeling we mean a pair one-to-one maps, one from V(G) to A, and another one from E(G) to A, rather than the bijection maps considered in the literature from V(G)∪E(G) to A, and new results were obtained.

Gaussian anti magic labeling in some special graphs

Consecutive magic graphs

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

Let A be a nontrivial abelian group and A* = A \ {0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to construct nontrivial classes of ℤp-magic graphs, prime p ≥ 3. For these graphs, some lower bounds on the number of distinct ℤp-magic labelings are also established.

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 to as k-magic graphs. In this paper we prove that the graphs such as star of cycle, flower, double wheel, shell, cylinder, gear, generalised Jahangir, lotus inside a circle, wheel, closed helm graph are Zk-magic graphs.

Умножение на абелевой группе G - это гомоморфизм $$\mu: G\otimes G\rightarrow G$$. Абелева группа G называется MT-группой, если любое умноженеие на ее периодической части однозначно продолжается до умножения на G. MT-группы изучались во многих работах по теории аддитивных групп колец, но вопрос об их строении остается открытым. В настоящней работе для MT-группы G рассматривается сервантная вполне характеристическая подгруппа $$G^*_\Lambda$$, одно из основных свойств которой заключается в том, что подгруппа $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ является ниль-идеалом в любом кольце с аддитивной группой G (здесь $$\Lambda(G)$$ - множество всех простых чисел p, для которых p-примарная компонента группы G отлична от нуля). Показано, что для любой MT-группы G либо $$G=G^*_\Lambda$$, либо факторгруппа $$G/G^*_\Lambda$$ несчетна.

The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.