Further Results on Product Cordial Labeling

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.

The product cordial labeling is a variant of cordial labeling. Here we investigate product cordial labelings for alternate triangular snake and alternate quadrilateral snake graphs.

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.

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 coloring of a graph \(G\) is called a majestic edge coloring if there is the induced proper vertex coloring. If the colors of edges of the \(G\) graph have 2-elements sets and the \(G\) graph has induced proper vertex coloring then an edge coloring of the \(G\) graph is called the majestic 2-tone coloring. The majestic and the majestic 2-tone chromatic indices for some cycle related graphs which are wheel graph, gear graph, helm graph, web graph, and friendship graph are computed.

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defi ned by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the graphs GL(n), double triangular snake D(Tn), Wn, Fln, and admit edge pair sum labeling.

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.

Signed Product Cordial Labeling for Some Families of Graphs

English

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.

On Signed Product Cordial Labeling of Pan-Related Graphs

On 3-total edge product cordial labeling of a carbon nanotube network

A graph with q edges is called antimagic if its edges can be labeled with 1, 2, 3, ..., q without repetition such that the sums of the labels of the edges incident to each vertex are distinct. In this paper we study antimagic labeling of double triangular snake, alternate triangular snake, double alternate triangular snake, quadrilateral snake, double quadrilateral snake, alternate quadrilateral snake, double alternate quadrilateral snake.

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).

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.