Cube difference labeling of some special graphs

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 .

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 $K_{m, n}$.

Gaussian anti magic labeling in some special 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.

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.

Let $G$ be a graph. Let $f:Vleft(Gright)rightarrow left{0,1,2,ldots,k-1right}$ be a function where $kin mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $fleft(uvright)=leftlceil frac{fleft(uright)+fleft(vright)}{2}rightrceil$. $f$ is called $k$-total mean cordial labeling of $G$ if $left|t_{mf}left(iright)-t_{mf}left(jright) right| leq 1$, for all $i,jinleft{0, 1, ldots, k-1right}$, where $t_{mf}left(xright)$ denotes the total number of vertices and edges labelled with $x$, $xinleft{0,1,2,ldots,k-1right}$. A graph with admit a $k$-total mean cordial labeling is called $k$-total mean cordial graph.

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.

Let G = (V (G), E(G)) be a graph with q edges. A function f is called harmonious labeling of graph G if f: V →¨{0, 1, 2,…, q-1} is injective and the induced function f*: E → {0, 1, 2,…, q} defined as f*(uv) = (f(u) + f(v))(mod q) is bijective. A graph which admits harmonious labeling is called harmonious graph. In this paper we prove that the jewel graph, triangular ladder graph, special flower graph, duplicating all the vertex of mK1, in P2+mK1, T(Pn)Kcm are harmonious graphs.

With the rapid development of wireless communication has made efficient spectrum assignment a crucial factor in enhancing network performance. As a combinatorial optimization model for channel assignment, the radio labeling is recognized as an NP-hard problem. Therefore, converting the spectrum assignment problem into the radio labeling of graphs and studying the radio labeling of specific graph classes is of great significance. For G, a radio labeling φ : V ( G ) → { 0 , 1 , 2 , … } is required to satisfy | φ ( u ) − φ ( v ) | ≥ diam ( G ) + 1 − d G ( u , v ) , where diam ( G ) and d G ( u , v ) are diameter and distance between u and v. For a radio labeling φ, its span is defined as the largest integer assigned by φ to the vertices of G; the radio labeling specifically denotes the labeling with the minimal span among possible radio labeling. The strong product is a crucial tool for constructing regular networks, and studying its radio labeling is necessary for the design of optimal channel assignment in wireless networks. Within this manuscript, we discuss the radio labeling of strong prismatic network with star, present the relevant theorems and examples, and propose a parallel algorithm to improve computational efficiency in large-scale network scenarios.