Abstract
A function is mentioned as a -exponential mean labeling of a graph that has vertices and edges if is injective and the generated function defined by for all , is bijective. A graph which recognizes a -exponential mean labeling is defined as-exponential mean graph. In the following study, we have studied the exponential meanness of the path, the graph triangular tree of , , cartesian product of two paths , one-sided step graph of , double-sided step graph of , one-sided arrow graph of , double-sided arrow graph of , and subdivision of ladder graph .
1. Introduction
In the field of mathematics, along with some areas of sciences, graph theory has become an interesting topic of study. A graph labeling is considered as an integer’s assignment to the edges or vertices, or vice versa, subjected to particular conditions. Many mathematicians and scientists have contributed and introduced different kinds of labeling [1–6].
In the present study, the graphs considered here are undirected, simple, and finite graphs that have vertices and edges. Referring to the graph labeling introduced by Gallain, a detailed survey is conducted on graph labeling [4]. Somasundaram and Ponraj [7] originated the theory of mean labeling of graphs. Many mathematicians introduced different aspects of mean labeling. The study of Kannan et al. on the exponential mean labeling of a few different graphs studied through duplicate operations is examined for the present study [8]. Barrientos’ study on alpha graphs has demonstrated the presence of α-labeling of a tree using various vertices and lengths of base path and proved that these trees can be utilized to demonstrate unicycle graphs with α-labeling [9]. Studies on cordial labeling between paths and cycles for a Cartesian product have demonstrated that these Cartesian products, under any conditions, are always cordial and even proved that two path Cartesian products are always cordial [10].
Sumathi and Rathi introduced the quotient labeling number for a wide family of ladder graphs, namely, closed triangular ladder, open triangular ladder, closed ladder, open ladder, step ladder, slanting ladder, and open diagonal ladder [11]. Baskar, referring to the flooring function edge labels, defined the logarithmic mean labeling on graphs and studied the logarithmic meanness of different ladder-related graphs [12].
Traditionally, the logarithmic mean of any two positive integers is not necessary to be an integer. And, if the logarithmic mean is considered an integer, the flooring of ceiling function is used. The edge label is set through flooring or a ceiling function, which is defined to be the logarithmic mean labeling of graphs. Baskar defined logarithmic mean labeling on graphs by setting the edge labels from flooring function [12]. A graph is considered a logarithmic mean graph if it recognizes logarithmic mean labeling. In 1967, Rosa proposed graceful labeling, known as β-valuation [13], and later, Golomb represented it as graceful labeling [1]. Kaneria et al., in 2010, introduced arrow graph and double arrow graph [14]. And, in 2015, step grid graph and double step grid graph were introduced [15]. These graphs were defined to be graceful graphs.
Motivated by such works, in this study, we aimed to work to introduce a new class of -exponential mean labeling for different ladder graphs, looking at the ceiling function. A graph which recognizes a -exponential mean labeling is defined as -exponential mean graph. In the present study, we have examined the exponential meanness of the path, the graph triangular tree of , , cartesian product of two paths , one-sided step graph of , double-sided steps graph of , one-sided arrow graph of , double-sided arrow graph of , and subdivision of ladder graph .
A function is mentioned as a -exponential mean labeling of a graph that possess vertices and edges if is injective and the generated function defined by for all , is bijective.
1.1. Preliminaries
The below-mentioned definitions are essential for the present study.
Definition 1. Let be the consecutive vertices of ; a triangular tree is calculated by amalgamating each with a leaf (or vertex of degree 1) of . We denote this tree by and refer to as the base of . Note that has size , which means that its order is a triangular number. We say that the first vertex of leaf is amalgamated with the vertex of
Definition 2. Let be a path on vertices represented by and with edges signified by , where represents the edge connecting and, the vertices. On every edge, erect a ladder that has steps counting the edge, for. The graph hence drawn is defined as a one-sided step graph, and it is represented by.
Definition 3. Let be a path on vertices with edges , where represents the edge connecting and , the vertices; on every edge , we erect a ladder that has steps counting the edge , for , and on every, erect a ladder that has steps counting, for . The graph hence drawn is defined as double-sided step graph, and it is represented by.
Definition 4. An arrow graph with breadth and length is acquired by joining a vertex with superior vertices of by new edges from one end.
Note. In the graph, (grid graph on mn vertices) vertices and vertices are known as superior vertices (Figure 1) from both ends.
Definition 5. A double arrow graph with breadth and length is calculated by joining a vertex with superior vertices of by new edges from both ends.
Definition 6. A graph, which can be formed from an identified graph G by dividing up each edge into exactly two segments by positioning intermediate vertices between its two ends, is called a subdivision graph. It is represented by.
2. Main Results
Theorem 8. Each triangular tree is a -exponential mean graph, for .
Proof. Assume denote the vertices of the path and each vertex adjoining the path that is represented by, for .
Define as follows:
Then, the generated edge labeling is calculated as follows:
Define as follows:
Hence, is a -exponential mean labeling of the triangular tree graph, for A typical example is illustrated in Figure 2.
Theorem 9. The graph is a -exponential mean graph, for, .
Proof. Assume denote the vertices of the cycle for. Then, path extended from the cycle through the vertices for
Define as follows:
Hence, the generated edge labeling is calculated as follows:
Define as follows:
when .
Hence, is a -exponential mean labeling of the graph, for Figure 3 depicts an example of the aforementioned labeling.
Theorem 10. The graph is a -exponential mean graph, for
Proof. The Cartesian product of the graphs and is the graph that has vertex set and edge set. In this article, the vertices of the graph are presented as a matrix with rows and columns. Moreover, we denote as the vertex which lies at the -th row and -th column where and .
Define as follows:
Then, the generated edge labeling is calculated as follows:
Define as follows:
Hence, is a -exponential mean labeling of the graph, for Figure 4 illustrates a representative example of the labeling described above.
Theorem 11. The graph is a -exponential mean graph, for
Proof. Assumedenote the vertices of the step graph.
Let be the step ladder graph with steps for
Let be the vertices on the base where
Let be the vertices on the second stage above the base for
Let be the vertices on the third step for
Proceeding like this, we have vertices for steps.
Now the vertices of is denoted by.
In , signifies the row (calculated through bottom to top) and signifies the column (calculated through left to right) in which the vertex occurs.
Now, the graph of vertices and edges are with ; for ; for ; for ; and for ,, and
But, and .
Define as follows:
Hence, the generated edge labeling is calculated as follows:
Define as follows:
Hence, is a -exponential mean labeling of the graph Thus, the graph is a -exponential graph for . A characteristic example of the labeling mentioned above is shown in Figure 5.
Theorem 12. The graph is a -exponential mean graph, for
Proof. Assume denote the vertices of the double-sided step graph. In, signifies the row (calculated through bottom to top) and signifies the column (calculated through left to right) in which the vertex occurs.
Define as follows:
The above-defined labeling pattern gives rise to as an injective map and defines as follows:
for all , is defined as bijective.
Hence, is a - exponential mean labeling of, Hence, the graph, for , is a -exponential mean graph for Figure 6 displays a distinctive illustration of the labeling stated before.
Theorem 13. The graph is a -exponential mean graph, where and
Proof. Assume is an arrow graph calculated by connecting a vertex with superior vertices of by two new edges.
Let be vertices of .
Join with by 2 new edges to obtain and .
by using -exponential mean labeling formula for all is defined as bijective.
The above-defined labeling pattern gives rise to as an injective map and defines as follows:
Hence, the graph is a -exponential mean graph, for and A sample example of the previously mentioned labeling is shown in Figure 7.
Theorem 14. The graph is a -exponential mean graph, where and
Proof. Assume is a double arrow graph calculated by joining two vertices with by four new edges on both sides.
Let be vertices of . Join with and with by four new edges to obtain. and .
Define as follows:
Then, the generated edge labeling is calculated as follows:
by using -exponential mean labeling formula, for all , is defined as bijective.
Hence, the graph is a -exponential mean graph, for Figure 8 displays an illustration of the labeling from earlier as an example.
Theorem 15. The subdivision of ladder graph is a -exponential mean graph, for
Proof. Assume The ladder graph is defined as , where is a path with signifing the cartesian product. Let be the ladder vertices . Let be the lately added vertex joining and, be the newly added vertex between and and be the lately added vertex joining and . Clearly, has vertices and edges.
Define as follows:
Hence, the generated edge labeling is calculated as follows:
Define as follows, for .
Hence, is a -exponential mean graph, for A representation of the prior labeling is shown as an example in Figure 9.
3. Conclusion
The -exponential mean labeling of tree, cycle-path, step graph, ladder graphs, arrow graphs, and subdivision of ladder graph was introduced and discussed in this work using graph operations.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
There is no conflict of interest to declare.
Acknowledgments
The authors gratefully acknowledge the Vellore Institute of Technology Management for providing support.