Abstract
Assume that is a connected graph. For a set of vertices , two edges are distinguished by a vertex , if . is termed edge metric generator for if any vertex of distinguishes every two arbitrarily distinct edges of graph . Furthermore, the edge metric dimension of , indicated by , is the cardinality of the smallest for . The edge metric dimensions of the dragon, kayak paddle, cycle with chord, generalized prism, and necklace graphs are calculated in this article.
1. Introduction and Preliminaries
A chemical compound’s structure is typically seen as a collection of functional groups arranged on a substructure. The structure is a labeled graph from a graph-theoretic outlook, with the vertex and edge labels indicating the atom and bond types, respectively. Functional groupings and substructure are essentially subgraphs of the labeled graph representation from this perspective. A collection of compounds described by the substructure common to them is fundamentally defined by modifying the set of functional groups and permuting their locations. Chartrand et al. defined chemistry as the application of graphs to illustrate the structure of diverse chemical compounds (see [1, 2]). We employ graph principles to explain chemical structures in chemical graph theory. Chemical compounds’ atomic structure can be presented using graphs. Johnson has worked on the application of structure-activity relationships by labeled graphs (see [3]).
By establishing the concept of metric dimension, Slater was able to locate the invader in a computer network (see [4]). Harary and Melter extended the work of Slater in [5]. Hosamani et al. studied the connection to chemical problems in [6]. Berhe and Wang calculated topological coindices for nanotubes and graphene sheets in [7]. Goyal et al. focused on the new composition of graphs in [8]. Ranjini et al. investigated the applications of molecular topology by degree sequence of graph operator in [9]. Imran et al. investigated chemistry problems in order to create mathematical representations of various chemical substances, with each compound having its own representation (see [10]). Navigation can be understood within a graphical structure in which the point robot or navigating agent moves from vertex to vertex of a graph. The robot can find its location by finding the distance from a fixed set of vertices also called landmarks. There is no idea of direction and visibility in a graph, but the point robot can find the distances from a fixed set of landmarks (see [11]). Melter and Tomescu worked on the metric distances used in image processing, for example, chessboard distance and the city block distance, and they also studied the applications to pattern recognition problems (see [12]). Raza investigated the chemistry application of polyphenyl and spiro chains (see [13]). Caceres et al. described the metric dimensions of graphs in coin weighing and mastermind games (see [14]). Liu et al. worked on the application of cellular neural networks in [15]. Some applications of graphs to chemistry, biology, and physics are discussed in [16, 17]. Ali et al. studied the irregularity of graphs in detail in [18].
Let be a vertex with degree and the total number of edges linking to be the degree of vertex . ’s maximum and minimum degrees are indicated by the symbols and , respectively. is known as the metric generator of if any two arbitrary elements of can be distinguished by some element in . The metric dimension of is the number of elements in the smallest .
Kelenc et al. [19] developed the concept of edge resolvability of graphs. The distance between edge and vertex is represented by , and it is defined as , where (see [19]). Edges and are distinguished by a vertex if . For every two distinct elements , there always exists such that. The edge basis of is thus known as the minimal , and is known as the edge metric generator. Furthermore, the minimum number of vertices in an edge basis is the edge metric dimension of , which is denoted by . A graph with only one cycle is referred to as a unicyclic graph, while a graph having more than one cycle is referred to as a multicyclic graph. We will look at both unicyclic and multicyclic graphs in this article.
Kelenc et al. in [19] introduced the new concept of and made its comparison with metric dimension. Zubrilina in [20] calculated the of graphs which has order . Filipovic et al. in [21] calculated the constant value of of graph for fixed values of and computed the lower bound for the rest of the values of . Ahsan et al. in [22] worked on the bounded and unbounded of some graphs. In addition, Ahsan et al. in [23] calculated the constant of two regular graphs. Mufti et al. in [24] computed the of Cayley graphs. Wei et al. calculated the of chordal ring network and H graph in [25]. The of various polyphenyl chains was determined by Ahsan et al. in [26]. Alrowaili et al. computed the of very important classes of Toeplitz graphs in [27]. Xing et al. compared the edge resolvability with the mixed metric of wheel graphs in [28]. Koam and Ahmad calculated the of barycentric subdivision of Cayley graphs in [29]. Peterin and Yero studied the of graph operations in [30]. Zhang and Gao calculated the of various graphs in [31]. Liu et al. developed the idea of fault of some graphs in [32]. Deng et al. computed the of different families of mobius networks in [33].
These lemmas are helpful in determining values of graphs.
Lemma 1 (see [19]). For any connected and simple graph ,(1).(2).
Lemma 2 (see [19]). For any ,,,. Furthermore, is path.
The article is structured as follows.
We will study the of the dragon graph , kayak paddle graph , cycle with chord graph , generalized prism graph , and necklace graph in Sections 2, 3, 4, 5, and 6, respectively. We will discuss two classes of graphs, first one is where the usual metric dimension is equal to the edge metric dimension and the second class is where the edge dimension is greater than the usual metric dimension. The graphs have symmetry according to cycles, first graph has one cycle, second graph has two cycles, third graph has three cycles, and last two graphs and have cycles.
2. Edge Metric Dimension of Dragon Graph
The will be computed in this section. The dragon graph has and . Figure 1 shows the graph for and . We will look at ’s metric dimension in the next theorem.
Theorem 1 (see [34]). For all ,,.
The following theorem computes the .
Theorem 2. For all , ,.
Proof. We must show that is an edge basis for graph if . We do this by computing each edge representation of . where , , and .
We see that any two tuples are not equal. This implies that and now we try to show that . Since by Lemma 2, is not a path, . As a result, .
3. Edge Metric Dimension of Kayak Paddle Graph
The will be computed in this section. The kayak paddle graph has and . Figure 2 shows the graph for n = 8, m = 5, and l = 4. The metric dimension of is presented in the next theorem.
Theorem 3 (see [35]). For , , and , .
The following theorem computes the .
Theorem 4. For , , and , .
Proof. Let , and we have to prove that is an edge basis for graph . We can do it by computing each edge representation of . where ., , , and .
We see that any two tuples are not equal. This implies that and now we try to prove that . By Lemma 2, . This proves that .
4. Edge Metric Dimension of Cycle with Chord Graph
The will be computed in this section. The cycle with chord graph has and . It suffices to consider . Figure 3 shows the graph for n = 16 and m = 8. We will look at ’s metric dimension in the next theorem.
Theorem 5 (see [35]). For all , .
The following theorem computes the .
Theorem 6. For all , .
Proof. To calculate the edge basis of , the following are the four case scenarios. Case (i). When both numbers and are even. With and and , we must show that is an edge basis for graph . We do this by computing each edge representation of . and . Case (ii). When the number is odd and the number is even. Let us say and , and let us take . We must show that is an edge basis for graph . We do this by computing each edge representation of . and . Case (iii). When is an even number and is an odd number. Assuming and , we must show that is an edge basis for graph . We do this by computing each edge representation of . and . Case (iv).When both numbers and are odd. Assuming and , we must show that is an edge basis for graph . We do this by computing each edge representation of . and .In all four case scenarios, we can observe that no two tuples have the same representations. This implies that . Because is not a path according to Lemma 2, . As a result, .
5. Edge Metric Dimension of Generalized Prism Graph
In this part, the will be calculated. The Cartesian product of path on vertices and cycle on vertices is the generalized prism graph . and in the generalized prism graph . Furthermore, is referred to as a prism graph. Figure 4 shows the graph for n = 4 and m = 6. We will look at ’s metric dimension in the next theorem.
Theorem 7 (see [14]). For all and , we have
Lemma 3. For and , .
Proof. Since , by Lemma 1, we have .
In the following theorem, the is determined.
Theorem 8. For all and , .
Proof. The following two case scenarios are used to compute the edge basis of . Case (i). When the number is even. Let , , where is a positive integer, and ; we must show that is an edge basis for graph . Each edge representation of is computed for this purpose. where . Case (ii). When is odd. Let , , where k is a positive integer and , and we have to prove that is an edge basis for graph . For this purpose, we compute each edge representation of . where .We see that any two tuples are not equal in both of the cases. This implies that . Now by Lemma 3, we have . Hence, .
6. Edge Metric Dimension of Necklace Graph
The will be computed in this section. The necklace graph has and . Figure 5 shows the family for . We will look at ’s metric dimension in the next theorem.
Theorem 9 (see [36]). For all , we have
Lemma 4. For , is the necklace graph family; then, .
Proof. Since , by Lemma 1, we have .
The is computed in the following theorem.
Theorem 10. For all, .
Proof. If , we must show that is an edge basis for graph . We do this by computing each edge representation of .We see that any two tuples are not equal. This implies that . Now we only prove that . So, by Lemma 4, we have . Hence, .
7. Conclusion
The edge metric dimension of the unicyclic and multicyclic graphs, dragon, kayak paddle, cycle with chord, generalized prism, and necklace, has been calculated in this paper. It should be underlined that the edge metric dimensions of all the described graphs are constant and do not rely on the number of vertices in the graph. The edge metric dimension of the dragon graph, kayak paddle graph, and cycle with chord graph is determined to be two, whereas the edge metric dimension of the generalized prism graph and necklace graph is determined to be three. We come to a halt here with an unsolved problem. The open problem is distinguishing between all forms of graphs in which the edge metric and metric dimensions are the same.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.