English

A graph G(V, E) admits an (a, d)-H-antimagic covering if every edge in E(G) belongs to H′ subgraph of G that is isomorphic to H and there exists a bijective function f : V (G) ∪ E(G) → 1, 2,…, |V (G)| + |E(G)| such that for all subgraphs H′ isomorphic to H, the H′-weights, w(H′) = ∑vεV (H′) f (v) + ∑eεE(H′) f(e) constitutes an arithmetic progression a, a + d,…, a + (t − 1)d, where a and d are some positive integers and t is the number of subgraphs isomorphic to H. If the label of vertices are {1, 2, …, |V (G)|}, then it is called super (a, d)-H-antimagic covering. In this paper we find super (a, d)-H-antimagic covering on double star graph Sn,n, union of star graph mSn and union of double star graph mSn,n.

The computational complexity of cordial and equitable labelling

Let G(V (G),E(G)) be a connected, simple, and finite graph. Let f be a bijective function of labeling on graph G from the edge set E(G) to natural number up to the number of edges of G. A rainbow vertex antimagic labeling of graph G is a function f under the condition all internal vertices of a path u – υ, Ɐu, υ ∈ V (G) have different weight (denoted by w(u)), where w(u) = ∑ uu′∈E(G)f (uu′). If G has a rainbow vertex antimagic labeling, then G is a rainbow vertex antimagic coloring, where the every vertex is assigned with the color w(u). The rvac(G) is a notation of rainbow vertex antimagic connection number of graph G which means the minimum colors taken over all rainbow vertex antimagic coloring induced by rainbow vertex antimagic labeling of graph G. The results of this research are the exact value of the rainbow vertex antimagic connection number of star (Sn ), double star (DSn ), and broom graph (Brn, m ).

In this paper, we introduce a new type of Labeling of a graph G (V, E) with |V(G)| vertices and |E(G)| edges as an injective map φ: V(G) → {1, … . k}, k ∊ ℕ such that the induced constant mapping φ*: E(G) → (0, 1) by φ*(e) = φ*(uv) = |φ(u) - φ(v)|p = λ, where λ ∊ (0, 1) called p-adic distance labeling, for any fixed prime ‘p’. Here, we prove the existence of this new p−adic distance labeling for some simple graphs such as Path (Pn), Cycle (Cn), Complete graph (Kn) and Star graph (K1, n), n ≥ 3 also for the Corona graph Pn⊙k2¯, when n≥2 and even and for p=2,3. And we explore the applications of this labeling in wireless networks.

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.

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.

The use of graph‐theoretic parameters is essential in interpreting the structural robustness, efficiency, and functionality of complex networks. Specifically, edge‐based metrics are more useful in interpreting the susceptibility of links and the concept of upstream impact, as well as structural robustness. Such metrics are essential in a variety of fields such as network security, communication, and optimization theory. In the same way, a new edge‐based invariant is introduced in this research paper, which is known as the b -edge differential . The b‐edge differential is essentially linked to b ‐differential of a graph. Let G = ( V , E ) be a simple connected graph. For any subset of edges X ⊆ E , we associate the set B ( X ) = { e ∈ E \ X : e shares a common vertex with at least one edge in X }, that is, B ( X ) captures all edges outside X which are adjacent to X . The quantity ∂ B E ( X ) = | B ( X )| is called the b ‐edge differential of X . We then define the b ‐edge differential of a graph G by ∂ B E ( G ) = max X ⊆ E ∂ B E ( X ). This parameter calculates the maximum value of the indirect dominances, or coverage, which can be achieved by a given set of edges. It serves as a useful tool in the identification of critical edges as well as the maximization of influence in networks. The problem can also be used in the applications involving the maximization of profit, traffic routing, and vulnerability evaluation for communication networks. We carry out the explicit computation of the b ‐edge differential for some classical types of graphs, such as paths, cyclic graphs, wheel graphs, complete graphs, star graphs, double‐star graphs, comb graphs, complete bipartite graphs, ladder graphs, and triangular ladder graphs. The work described in this article expands the relevance of differential parameters in the theory of graphs, with promising research directions for theoretical and application aspects of network science.

Let G = (V,E) be a graph with vertex set V and edge set E. The graph G is said to be a local irregular vertex coloring if there is a function f is a called a local irregularity vertex coloring if : (i) l : (V (G)) → {1, 2,…,k} as a vertex irregular k-labeling and w : V(G) → N, for every uv ∈ E(G),w(u) = w(v) where w(u) = Σv∈N(u) l(i) and (ii) opt (I) = min{rnax{li; livertex irregular labeling}}. The chromatic number of local irregularity vertex coloring of G denoted by Xlis(G), is the minimum cardinality of the largest label over all such local irregularity vertex coloring. In this paper, we study local irregular vertex coloring of G ⊳∘ Sn when G is complete bipartite graph (Km,p), ladder graph (Lm), fan graph (Fm), friendship graph (Frm), and cycle graph (Cm).

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.

In this paper, we introduce a coding technique for converting a text message using sub super mean labeling on two and three star graphs to a picture coding message. The first part of this paper there are some definitions and discussions for the existence graph and in the second part there are some illustrations to explain how the text message is converted in to picture code, by using graph labeling, and it maintain secrecy, possess intricacy and provide a sense of sufficiency. Secrecy is the practice of hiding information from certain individuals or groups who do not have the need to know, perhaps while sharing it with other individuals that which is kept hidden is known as secret.

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.

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.

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.

Let G be a graph which has V(G) as a vertex set and E(G) as an edge set. The G graph contain subgraph H that isomorphic with Hj, j = 1, 2, … s, if every e ∈ E(G) include only in one of the edge set of H subgraph (e ∈ E(H)). The total l-labeling is a graph labeling that give label positive integer number until l into vertices and edges. The total H-irregular labeling is a total l-labeling with condition that the sum of vertex labels and edge labels in two distinct subgraphs H 1 and H 2 isomorphic to H is different. We define H-weight as wtφ (H) = ∑ v∈V(H) φ(v) + ∑ e∈E(H) φ(e), for the subgraph H ⊆ G under the total l-labeling (φ). The total H-irregularity strength of G graph (ths(G, H)) is the smallest l value in label of G graph has total H-irregular labeling. We used plane graphs such as triangular ladder graph and windmill graph.

A (0; 1)-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let g be a labeling of the edge set of a graph that is induced by a labeling f of the vertex set. If both g and f are friendly then f is said to be a cordial labeling of the graph. This concept extended to directed graphs is called (2; 3)-cordiality of digraphs. We investigate the labelings that are both cordial for a graph and (2; 3)-cordial for an orientation of it. We also consider the same problem for other known binary vertex labelings of graphs.