Product cordial labeling for alternate snake graphs

In this paper we consider the cordiality of a generalized Jahangir graph $J_{n,m}$. We give sufficient condition for $J_{n,m}$ to admit (or not admit) the prime cordial labeling, product cordial labeling and total product cordial labeling.

An injective function f : V (G) → {T0, T1, T2, . . . , Tn}, where n = |V (G)| − 1, is said to be a Tribonacci cordial labeling if the induced function f ∗ E(G) →{0, 1} defined by f∗(uv) = (f(u) + f(v)) mod 2 satisfies the condition |ef (0) − ef (1)| ≤ 1 where ef (0) is the number of edges with label 0 and ef (1) is the number of edges with label 1. A graph that admits a tribonacci cordial labeling is called a Tribonacci cordial graph. In this paper, we determined the Tribonacci Cordial Labeling of Triangular Snake Graph T Sn, Double Triangular Snake Graph D(T Sn), Quadrilateral Snake Graph QSn, Double Quadrilateral Snake Graph D(QSn), and Cycle Quadrilateral Snake Graph C(QSn).

The computational complexity of cordial and equitable labelling

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.

Let G = (V,E) be a simple graph. L(2, 1)−labeling defined as a functionf : V (G) → N0 such that, x and y are two adjacent vertices in V, then if x andy are adjacent to each other, |f(y) − f(x)| ≥ 2 and if x and y have the distance 2,|f(y) − f(x)| ≥ 1. The L(2, 1)-labeling number of G, called λ2,1(G), is the smallestnumbermof G. In this paper, we will further discuss the L(2, 1)-labeling of mongoliantent, lobster, triangular snake, and kayak paddle.Keywords: L(2,1)-Labeling, mongolian tent, lobster, triangular snake, kayak paddle.

<p>We prove that closed helm CHn, web graph Wbn, flower graph Fln, double<br /> triangular snake DTn and gear graph Gn admit product cordial labeling.</p>

<p>The concepts of cordial labeling, signed product cordiality, and logical cordiality have been introduced independently by different researchers as distinct labeling schemes. In this paper, we demonstrate the equivalence of these concepts. Specifically, we prove that a graph $ G $ is cordial if and only if it is signed product cordial, if and only if it is logically cordial. Additionally, we establish that a graph $ G $ admits permuted cordial labeling if and only if it exhibits cubic roots cordial labeling. Furthermore, we leverage this newfound equivalence to analyze the cordiality properties of several standard graphs.</p>

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.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.

An edge odd graceful labeling of graph G with q number of edges is a bijection f : E(G) → {1,3,5, …,2q - 1} so that induced mapping f+ : V(G) → {0, 1,2, … ,2q - 1} given by f+(x) = ExyEE(G) f(xy) (mod 2q) is injective. A triangular snake net graph (m - 1)C3m is graph obtained from a path u1 u2 u3 … um + 1 by joining every ui and ui + 1 to a new vertex vi with 1 ≤ i ≤ m and joining every vi to vi+1 with 1 ≤ i ≤ m - 1. A quadrilateral snake net graph C4,4m is a graph obtained from vertices u1, u2, u3, …, u2m + 1 by joining every ui and ui + 1 to two vertices vi and v2i-1with 1 ≤ i ≤ 2m, and joining every v4i - 2 and v4i to a new vertex w2i and joining every v4i - 3 and v4i-1 to a new vertex w2i - 1 with 1 ≤ i ≤ m. An alternate quadrilateral snake net A(C4,4m) is graph obtained from vertices u1, u2, u3, … , u3m by joining every u3i and u3i + 1 with 1 ≤ i ≤ m - 1, and joining u3i - 2 and u3i - 1 to two new vertices v4i-3 and v4i-2, joining u3i-1 and u3i to two new vertices v4i-1 and v4i, joining v4i-3 and v4i-1 to a new vertex w2i-1, and joining v4i-2 and v4i to two a new vertex w2i with 1 ≤ i ≤ m. A quadrilateral pleated snake graph C42 is a graph obtained from vertices u0, u, and v by joining every u0 and u to new vertices ui and joining every u0 and v to new vertices vi with 1 ≤ i ≤ 2 m + 2. In this paper, we study edge odd graceful labeling for the following families of snake graphs such as triangular snake net, quadrilateral snake net, alternate quadrilateral snake net, and quadrilateral pleated snake.

The study of digraphs provides a proving ground where mathematicians’ ability to bind together multiple disciplines of mathematics becomes evident. The new class of graph called Arithmetic graph was introduced on the basis of Number theory, particularly the Theory of Congruence. Graham and Spencer brought forth the idea of using quadratic residues to construct a tournament with p vertices where p ≡ 3 (mod 4) is a prime. These tournaments were appropriately named Paley digraphs in honor of the late Raymond Paley, who used quadratic residues 38 years earlier to construct Hadamard matrices. In this paper we prove that quadratic residue digraphs over a finite field called Paley tournament allows Edge product cordial labeling, K edge graceful labeling and Hn cordial labeling.

Sholairaju and Chithra introduced a graph which admits an edge odd graceful labeling called an edge odd graceful graph. An edge odd graceful labeling of a graph G on q edges is a bijection f : E(G) → {1,3,5, …, 2q-1} so that the induced mapping f+: V(G) → {0,1,2, …,2q - 1} given by f+(x) = Σxy∈E(G) f(xy) (mod 2q) is injective. A triangular snake C3m is a graph obtained from a path u1u2u3 … um+1 by joining every ui and ui+1 to a new vertex vi. A quadrilateral snake C4m is a graph obtained from vertices u1, u2, u3, …, um+1 by joining every ui and ui+1 to two vertices vi and wi. In this pa r we study edge odd graceful labelings of triangular snake C3m and quadrilateral snake C4m. We find a labeling of C3m and a labeling of C4m. This implies that graphs C3m and C4m are odd graceful, and we have new classes of graphs that are odd graceful.

This study investigates the application of Smarandache product cordial labeling techniques to Christmas tree graphs. Graph labeling is a crucial aspect of graph theory, with applications in various fields such as computer science, coding theory, and network design. Smarandache product cordial labeling, in particular, offers a unique approach to labeling graphs while maintaining specific properties. By exploring the application of these techniques to Christmas tree graphs, this research aims to contribute to the understanding of graph labeling and its potential applications in real-world problems. The study presents theoretical results, algorithmic approaches, and potential extensions to related graph structures.

The line graph L(G) of a graph G is the graph whose vertex set is E(G) and two vertices are adjacent in L(G) whenever they are incident in G. In this paper we have investigated product cordial labeling of line graph of some product cordial graphs.