Prime Labeling of Bull Graph

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

Let G be a simple graph of order n. Prime labeling is a bijective function f:V(G)→{1,2,…,n} such that gcd⁡(f(u),f(v))=1 for every pair of adjacent vertices u,v in G. A graph G that satisfies the definition of prime labeling is called a prime graph. The Dutch windmill graph D_r^n is a graph obtained by taking n copies of cycle graph C_r with a vertex in common. The double quadrilateral graph DQ is a graph constructed from two copies of C_4 and identifying one edge from each of them. The graph obtained by taking n copies of DQ and identifying one vertex of degree 3 from each of them as a common central vertex is called the double quadrilateral Dutch windmill graph DQ_n, for n≥1. Furthermore, graphs D_r^n and DQ_n becomes the base graph to construct two new graph classes, namely graph P_2 [D_r^n] and flower double quadrilateral graph FDQ_n. Both graph classes, constructed from the Dutch windmill graph, also contain even cycles. From previous research, it is known that graphs P_2 [D_4^n] and flower double quadrilateral graph FDQ_n have odd harmonious labeling. However, the determination of prime labeling on both classes is still an open problem. In this paper, we show that two classes of graphs constructed from Dutch windmill graphs with even cycles, namely graphs P_2 [D_4^n] and flower double quadrilateral graphs FDQ_n for n≥1 have a prime labeling. The result of this research shows that these graphs are prime graphs.

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

A graph H is a prime distance graph if there exists an one-to-one function h : V (H ) \(\to\) Z such that for any two adjacent vertices x and y, the integer |h(x) - h(y)| is a prime. So H is a prime distance graph if and only if there exists a prime distance labeling of H. If the edge labels of H are also distinct, then h becomes a distinct prime distance labeling of H and H is called a distinct prime distance graph. The generalized Petersen graphs P (n, k) are defined to be a graph on 2n (n \(\ge\) 3) vertices with V (P (n, k)) = {vi , ui : 0 \(\le\) i \(\le\)n - 1} and E(P (n, k)) = {vi vi+1 , vi ui , ui ui+k : 0 \(\le\) i \(\le\) n - 1, subscripts modulo n}. In this chapter, we give a concise overview on these two labeling, specifically, highlight that the generalized Petersen graphs P (n, 3) permit a prime distance labeling for all even n > 5 and conjecture that P (n, 2) and P (n, 3) admit a prime distance labeling for any n \(\ge\) 5 and all odd n \(\ge\) 5, respectively. Moreover, the cycle Cn admits a distinct prime distance labeling for all n \(\ge\) 3.

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.

kth Fibonacci Prime Labeling is defined as labeling the vertices of a graph with distinct Fibonacci numbers starting since the kth Fibonacci term sustaining the condition that the , where and are labels of any adjacent vertices u and v. Graphs formed by consecutively connecting identical base graphs, linearly or in alternating pattern, is called Snake graph. In this paper, we show that some snake graphs admit kth Fibonacci prime labeling.

We discuss vertex prime labeling of graphs obtained by using Shel graph. We have obtained vpl of Shel graphs, antena of Shel, Path union of Sn (n =4), snake of Shel and one point union of Shel, star of shel and a few other.

In this paper, we discuss about the prime labeling of Split graph of Coconut tree CT(m,n).Here m is the number of vertices in the path and n is the number of pendent vertices in the Coconut tree.A graph G with vertex set V is said to have a prime labeling if its vertices are labeled with integer 1,2,….,|V|such that for every edge xy the labels assigned to x and y are relatively prime.A graph which admits prime labeling is called a prime graph.

Graph labeling is the term used to describe the procedure of giving unique identifiers. Graph attributing is a method for modeling real-world problems in industries such as circuit design, surveillance technology, and telecommunications networks. A graph is characterized as possessing cubic roots cordial labeling if each node is designated with either the value 1, δ or δ2 or another specified integer. The derived set containing the value 1 or another integer, the mathematical conditions η (k) − η (w)| ≤ 1 and | ξ (k) − ξ (w)| ≤ 1, k ≠ w and k, w ∈ {1, δ, δ2} where η (k) and ξ (k) are represents the aggregated vertices and edges assigned in k ∈ {1, δ, δ2}. Labeled graphs prove to be immensely beneficial across a variety of mathematical frameworks, including domains such as cyber security and radar code formulation. The application of cordial labeling has been demonstrated to provide significant advantages in the context of signal processing channels and challenges associated with specific code word design. The idea of cubic roots cordial labeling has attracted a lot of academic attention among various labeling techniques, unique structural characteristics and capacity to depict a fair flow of information within linked networks. The investigation of the behavior of cubic roots cordial labeling in relation to graph operations is currently receiving extensive scholarly examination, particularly concerning path graphs, despite its critical relevance. This paper explore the cordial labeling of cubic product graphs generated from the combination of two path graphs. The four fundamental graph operations on which we particularly focus on activities, we analyze the specific conditions to possess cubic roots cordial labeling.

L(2,1)-colorings and Irreducible No-hole Colorings of Cartesian Product of Graphs

Group $S_{3}$ cordial prime labeling of graphs

This study investigates the existence of discrete and strong discrete labeling for several classes of path and star-related graphs, contributing to the broader field of graph labeling theory. A graph is said to admit a discrete labeling if its vertices can be assigned binary labels (0 and 1), such that each edge receives a label determined by the exclusive OR (XOR) of the labels of its incident vertices, subject to specified constraints. We first establish that the super subdivision of a path <img src=image/13441705_01.gif> by the complete bipartite graph <img src=image/13441705_02.gif> is discrete, demonstrating that replacing each edge of <img src=image/13441705_01.gif> with <img src=image/13441705_02.gif> preserves the discrete labeling property. Additionally, we prove that the square of a path <img src=image/13441705_01.gif>, obtained by connecting vertices at a distance of at most two, also admits a discrete labeling. Further, we examine the shadow graph <img src=image/13441705_04.gif>, constructed by duplicating <img src=image/13441705_01.gif> and connecting each vertex to its copy's neighbors, and show that it is also discrete. Another variant, the graph <img src=image/13441705_03.gif> formed by switching one pendant vertex in <img src=image/13441705_01.gif>, is similarly proven to support discrete labeling. Beyond paths, we explore star-related graphs, confirming that the shadow graph of a star <img src=image/13441705_05.gif> and the shadow graph of a bi-star <img src=image/13441705_06.gif> both permit discrete labeling. Extending the concept, we introduce strong discrete labeling, which imposes the stricter requirement by adding one more constraint to discrete labeling. Under this stronger condition, we prove that the star graph <img src=image/13441705_07.gif> admits a strong discrete labeling if and only if m is even, highlighting a parity-based constraint absent in standard discrete labeling. These findings expand the known families of graphs with discrete and strong discrete labeling, offering new insights into structural properties that facilitate such labeling. Potential applications include coding theory, network design, and algorithm optimization, while open questions remain regarding the extensibility of these results to other graph operations and more complex graph classes.

This paper discusses fuzzy labeling graph and its properties. The properties discussed are fuzzy labeling subgraph, union graph, and fuzzy magic graph. The results showed that the strength of connectedness for pair of vertices in a graph with fuzzy labeling is always greater than or equal to the strength of connectedness for pair of vertices in its subgraph. In addition, union of two graph with fuzzy labeling will form a graph with fuzzy labeling. Meanwhile, on magic labeling there is always a fuzzy bridge on fuzzy magic graph.