Abstract
This article comprises of exact valuation of a graph parameter, known as the edge irregularity strength , symbolized as , of various graphical families such as middle graph of path graph, middle graph of cycle graph, snake graph (string 2), paramedian ladder, and complete -partite graphs. If is a function defined on vertices of a graph that helps to determine different weights for every pair of edges, the least value of is the target. Thus, addition operation for allocated to vertices of an edge, i.e., , , defines the weight of corresponding edge for every . If two different edges and in graph carry weights in different manner, i.e., for . Then the edge irregular -labeling is defined after a vertex -labeling of . After establishing various novel results and making some conclusions, an open problem is mentioned in the end.
1. Introduction
In the modern era, several researchers have shown interest in graph labeling, and so many label findings emerge every year. Such success is attributed not just to the technical complexities of graph labeling but also to the diverse range of applications such as electron microscopy, X-ray, combinatorial optimization, sonar, physics, signal conditioning, network programming, and system layout.
Undirected, simple, and finite graphs are being considered in this article, for instance, a graph structure with pair of sets (vertices) and (edges/nodes). Then a mapping that assigns non-negative values to some graph elements is referred to as labeling of . If a mapping uses all vertices as well as edges of , then it is known as total labeling. While for a domain containing or only, then the associated labeling is vertex-labeling or edge-labeling in a respective manner. Therefore, in order to have an edge -labeling, , the corresponding vertex carries a weight equal to the labels sum of edges , , where are the adjacent vertices to i.e., . Other domains are possible, harmonious, cordial, graceful, magic and super magic are examples of graph labeling.
Baca and Siddiqui [1] defined the edge -labeling from to such that for all of a graph . The least value is the irregularity strength, and the highest label of graph . This variable has attracted much attention of researchers [2–8].
In 2012, Nurdin [9] considered the banana and quad trees in order to determine their total vertex irregularity strength . Furthermore, Ahmad et al. [4] took the disjoint union of Halm graph and evaluated the . Some results relating to graph labeling can be found in [1, 6, 10–13].
Al-Mushayt et al. [6] evaluated the total vertex inconsistency quality (irregularity strength) of a graph named convex polytope in 2013. A mapping is said to be edge irregular -labeling of if for all there exist , where the and . The minimal among upper bound of range set for which the graph has an edge irregular -labeling is said to be edge strength of denoted by . In 2014, Ahmed et al. [14] evaluated the bounds for and decided careful estimation of for paths, stars, twofold stars, and Cartesian result of two paths.
In this paper, we aim to label the graphs in an irregular manner and a few families in the relation of graphs, namely, middle graph of path graph, middle graph of cycle graph, snake graph (string 2), and paramedian ladder, and complete -partite graphs are considered to estimate their irregularity strengths.
Following are some important results developed in literature of graph theory [14].
1.1. Theorem A
For any simple graph whose degree as largest, then max .
1.2. Theorem B
For be a path on vertices, .
1.3. Theorem C
For be a star on vertices, .
1.4. Theorem D
For be twofold star with , .
1.5. Theorem E
Edge inconsistency quality of Cartesian product of two paths and is with .
1.6. Theorem F
is the order of graph. is the sequence of Fibonacci numbers obtained by the narration with with initial values and . Then of graph is less than or equal to .
2. Main Results
This section defines the middle graph of the path and the cycle graphs. Irregular labeling will be formulated and a lower bound of irregularity for different graphs will be found.
2.1. Middle Graph of Path Graph
If vertices carry a path , and is another set of vertices. Then the graph is said to be middle graph of path graph including the sets of vertices and edges, respectively, given as follows:
2.2. Middle Graph of Cycle Graph
If is cycle of vertices and is another set of vertices. Then the graph is said to be middle graph of cycle graph including the sets of vertices and edges, respectively, given as follows:
2.3. Ceiling Function
We define ceiling function as , when image of any value is in fraction like 5/2, we take this value as 3.
Theorem 1. Let is middle graph of the path graph with . Then .
Proof. Consider be path graph of vertices, is middle graph of the path graph including the following sets, and , of vertices and edges, respectively, given as follows:Order of graph is and size is . Presently, we characterize a mapping as follows:Degree of each is 4 and degree of , , and is 2. No two vertices have same labels for irregular labeling. The label of vertex is which is edge strength of . Weights of edges are given as follows:Edge weights are distinct for all distinct edges, i.e., for any two edges and . Thus apex (vertex) labeling is an edge irregular labeling. Hence, we have .
Example 1. The diagram in Figure 1 is the example of irregular labeling of that carries .
Theorem 2. Let is Cycle graph, is middle graph of cycle graph. Then
Proof. Let be a cycle graph of vertices and is middle graph including and as vertex set and edge set, respectively, given as follows:The order of is and size is , while the degree of each is 4 and the degree of each is 2 and no two vertices have same labels for irregular labeling. In this graph there are two possible cases. Case-i: when is odd we define function as follows: Weights of edges are as follows: Here distinct edges carry distinct weights, i.e., for any two edges and . Last vertex of has label . Thus, vertex labeling is an edge irregular labeling. Hence . Case-ii: for even we define transformation as follows:Weights of edges in this case are given as follows:Here the edge weights are distinct for all pairs of distinct edges i.e., for any two edges and , and last vertex of has label . Therefore, labeling is an edge irregular labeling. Hence, we have .
Example 2. The diagram in Figure 2 is the example of irregular labeling of that carries .
Theorem 3. is path with vertices and is complete bipartite graph. Graph is paramedian ladder graph. Then
Proof. let and are the edge and vertex sets, respectively, of the graph .The order of G is and size is . From Theorem A, . represents degree of the graph. Since degree of is so . To show equality it suffices to prove existence of an edge irregular labeling. For this we define mapping as follows:Weights of edges are given as follows:Weights form (2,1) irregular labeling. For any two different edges, their loads are different. Thus vertex labeling is an edge irregular labeling. Hence, we get .
Example 3. The diagram in Figure 3 is the example of irregular labeling of that carries .
Theorem 4. Theorem 4.Let be a snake graph of -rectangles with string . Then .
Proof. Let snake graph with -rectangles includes following sets and of vertices and edges, respectively,The order of is and size is . From Theorem A, , . Since max degree of is 4 so . To show equality, it suffices to prove existence of an edge irregular labeling. For this we define mapping as follows:Last vertex of has label . Furthermore, the weights of edges are given byEdge weights are distinct for all pairs of distinct edges i.e., for any two edges and . Thus, vertex labeling is an edge irregular labeling. Hence, we have .
Example 4. The diagram in Figure 4 is the example of irregular labeling of a snake graph of 5-strings and carries .
Theorem 5. Let be a m-partite complete graph with m set of vertices and each set contains n vertices. Then for m = 3, .
Proof. let and denote the vertex set and edge set of , respectively,The order of the graph is and size is . Since . Max degree of is . Now we define function such thatBy definition of labeling, label of each vertex must be different i.e., , and , otherwise, if for some and , then i.e., . Last vertex of has label . Further weights of edge are given as follows:Edge weights are distinct for all pairs of distinct edges, i.e., for any two edges and . Thus, the vertex labeling is an edge irregular labeling. Hence, we get . Furthermore, we can have and .
Example 5. The diagram in Figure 5 is the example of irregular labeling of that carries .
3. Conclusion
In this manuscript, we have examined the labeling of some families of graphs in an irregular manner and also introduced as a modification of different strengths like and a total . We got a few estimations on the and find precise incentives for certain families of graphs namely the middle graph of path, middle graph of cycle, snake graph, string 2, paramedian ladder, and complete -partite graphs. The value of can help for calculate the spanning trees for these type of graph. The graph labeling has wide range of applications such as x-rays crystallography, coding theory, radar, astronomy, and communication design. By raising the open problem, we conclude this section [14, 15].
3.1. Open Problem 1
For any positive integers , generalization of edge irregularity strength of -partite graph is , i.e., for .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors appreciated the very constructive comments from anonymous reviewers that have helped a lot to enhance the value of this manuscript. This study does not carry a funding from any financial body.