Abstract
This study introduces the average assignments and investigates its properties using various ladder graphs. The ladder graphs can be found in every communication networks. Ladder networks are increasingly being used in everyday life for monitoring and environmental applications such as domestic, military, surveillance, industrial, medical applications, and traffic management. These datasets are afflicted by the average representation of the graph structure. It aids in the visualisation and comprehension of data analysis. The labeling is used in sensor networks, adhoc networks, and other applications. It also efficiently creates a communication network after using noise reduction methods to remove salt and pepper noise.
1. Introduction
A graph labeling is the assignment of labels, conventionally indicated by integers, to edges and/or vertices of a graph in the mathematical domain of graph theory. The concept of labeling may be applied to many areas of graph theory, for example, in automata theory and formal language theory. We use [1–5] for notations and nomenclature. We recommend [6] for a thorough examination of graph labeling. Let be a path on nodes denoted by , where , and with lines denoted by , where , where is the line joining the vertices and . On each edge , erect a ladder with steps including the edge , for . The resulting graph is called the one-sided step graph, and it is denoted by . Let and be any two graphs with and vertices, respectively. Then, is the Cartesian product of two graphs. A ladder graph is the graph . The graph is obtained from by attaching pendant vertices to each vertex of . The triangular ladder , for , is a graph obtained from two paths by and by adding the edges and . The slanting ladder is a graph obtained from two paths and by joining each , with . The graph having the vertices and its edge set is .
2. Literature Survey
In [7], the authors talked about the -root square mean labeling for line graph of the path, cycle, star, , ladder, slanting ladder, the crown graph , and the arbitrary subdivision of . The authors in [8] discussed -face mean labeling some planar graphs and, in [9], face labelings of type for generalized prism. Alanazi et al. explained the classical meanness of the graphs, the one-sided step graph , double-sided step graph , planar grid , ladder graph , graph for , triangular ladder graph , graph for , graph for , slanting ladder graph , graph for , graph , diamond ladder graph , and latitude ladder graph in [10–12]. Dafik slamin et al. highlighted the super -edge-antimagic total properties of triangular book and diamond ladder graphs in [13]. Moussa and Badr discussed the odd gracefulness of few ladder graphs and proved that ladder and subdivision of ladder graphs with pendent edges are odd graceful in [14]. In [15], the authors emphasized the significance of exponential mean labeling of graphs, and they examined the exponential mean labeling of some graphs obtained from duplicating operations. Inspired by such tremendous works of researchers in the region of graph assignments in [16–25], we defined average assignment of graphs. A average of two integers is not always an integer. Consequently, average assignment must be an integer; we may get ceiling function by considering the integral part. In this study, our conversation and attempt is to examine the various assignment techniques on average assignment for few ladder graphs.
3. Methodology
A function is known as an average assignment of if is one to one and the instigated bijective function characterized bywhere , , is the number of edges, and is the set of all natural numbers. A graph that concedes an average assignment is known as a average assignment graph.
Figure 1 shows the average assignment of the graph .
4. Main Results
Theorem 1. The one-sided step graph is an average assignment graph, for .
Proof. Make the vertex assignment , .
Consequently, the instigated edge assignment is acquired as follows:As a result, for , is an average assignment and the one-sided step graph is an average assignment graph.
Theorem 2. The graph is an average assignment graph, for and .
Proof. Case (i): . Make the vertex assignment, . . Consequently, the instigated edge assignment is acquired as follows. . . Case (ii): . Make the vertex assignment, . . Consequently, the instigated edge assignment is acquired as follows: Case (iii): and . Make the vertex assignment: Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the graph is an average assignment graph, for .
Corollary 1. Every Ladder graph is an average assignment graph.
Theorem 3. The graph is an average assignment graph, for and .
Proof. Case (i): . Make the vertex assignment: , Consequently, the instigated edge assignment is acquired as follows: Case (ii): . Make the vertex assignment: , Consequently, the instigated edge assignment is acquired as follows: As a result, is an average assignment and the graph is an average assignment graph, for and .
Theorem 4. The triangular ladder graph is an average assignment graph, for .
Proof. Make the vertex assignment; ,Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the triangular ladder graph is an average assignment graph, for .
Theorem 5. The graph is an average assignment graph, for and .
Proof. Case (i): . Make the vertex assignment; , Consequently, the instigated edge assignment is acquired as follows: Case (ii): . Make the vertex assignment; , Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the graph is an average assignment graph, for and .
Theorem 6. The slanting ladder graph is an average assignment graph, for .
Proof. Make the vertex assignment; ,Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the graph is an average assignment graph.
Theorem 7. The graph is an average assignment graph, for and .
Proof. Case i: and . Make the vertex assignment, . Consequently, the instigated edge assignment is acquired as follows: Case (ii): and . Make the vertex assignment; , Consequently, the instigated edge assignment is acquired as follows: Case (iii): and . An average assignment of the graphs and are shown in Figures 1 and 2.As a result, is an average assignment and the graph is an average assignment graph, for and .
Theorem 8. The graph is an average assignment graph, for .
Proof. Make the vertex assignment; ,Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the graph is an average assignment graph, for .
Theorem 9. The diamond ladder graph is an average assignment graph, for .
Proof. Make the vertex assignment; ,Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the diamond ladder graph is an average assignment graph, for .
Theorem 10. The latitude graph is an average assignment graph.
Proof. Make the vertex assignment; ,Consequently, the instigated edge assignment is acquired as follows:As a result, is an average assignment and the latitude graph is an average assignment graph.
5. Conclusion
The significant properties on average assignment of several ladder networks are discovered in this work. Analysis of the average assignment of other networks can be discussed further.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.