#### Abstract

If the degrees of any two consecutive vertices differ by exactly one, the graph is called a stepwise irregular graph. The study of choosing specific vertices in an ordered subset of the vertex set such that no two vertices have the same representations with regard to the chosen subset is known as resolving parameters. This concept has been expanded for edges as well as a combined version of both. We examine a unique unicyclic stepwise irregular graph and an extended structure of stepwise irregular graph in terms of resolvability parameters to connect the stepwise irregular graph and resolving parameter concepts.

#### 1. Introduction

Let be a simple, connected, undirected graph with as the vertex set and as the edge set. The degree of a vertex , which is the count of edges related to , is denoted by . If every vertex in has the same , the graph belongs to a regular class of graphs; otherwise, it is irregular. Regular graphs have a history that may be traced back to either theoretical or practical applications.

In 1988, a question was aroused by Chartrand et al. [1] that “what if the graph vertices do not have the same degree? In which class of graphs it will be considered?” Although in 1987 the idea of highly irregular graphs was posed in [2], this idea begins with the new thoughts of studying graphs. To answer the question that “how irregular a graph is?”, the Albertson index [3] was introduced, which is defined as follows:where is the imbalance of an edge . The updated version of this index is the Mostar index [4]. Motivated by the Albertson index, Gutman in 2018 proposed a new idea which is known as a stepwise irregular graph and is defined as for each [5]. To count the number of cycles in a graph, the cyclomatic number of a graph is [5].

Researchers of [6] proposed the concept of metric dimension, which was based on the idea of identifying the location of invaders in networks first and foremost and called the metric generators as locating sets. The idea of metric dimension for trees is defined in [6] and for any graph in [7]. The resolving sets are also known as metric generators, using a similar technique. Radar or loran terminals were classified as components of the metric base set. If a single station fails to work, then the entire set resulted in failures to serve as a station hub. To overcome this problem, in 2008, the authors of [8] proposed an idea of a fault-tolerant resolving set. The idea of the adjacency generator was established in [9]; it was introduced to explain the metric dimension of composition of two networks.

Putting natural thinking on the pattern of recognizing the vertices of a graph by a subset (resolving set) of the vertex set, the edge metric dimension defined in [10], and later analogous to this definition, the authors of [11] proposed a mixed metric dimension in which both concepts (recognising vertices and edges) are combined, with the additional condition that no two elements (vertices and edges) have the same distances with respect to a chosen subset.

For , the distance (also called as geodesics) between vertices and is the count of edges between and . Let be a vertex for distinguished two vertices and , if . A set is called a resolving set for , if any pair of distinct vertices of is uniquely determined by some vertices of . The minimum cardinality of is named as a resolving set, and its cardinality is the metric dimension of , denoted by . If for each , remains a resolving set for , then is named as a fault-tolerant resolving set and the minimum count of elements of is called as a fault-tolerant metric dimension. For a vertex and an edge , the distance between and is measured as . A vertex identifies two edges , if . A subset having minimum vertices from a graph is an edge resolving set for , if each pair of two distinct edges of are distinguished by some vertex of . The minimum cardinality of an edge resolving set for a graph is called the edge metric dimension and is denoted by . A vertex of a graph distinguishes two elements (vertices or edges) of , if . A subset is a mixed resolving set, if any vertex and edge of are distinguished by some vertex of . The smallest cardinality of a mixed resolving set for is called the mixed metric dimension and is denoted by . Moreover, the mixed metric dimension is a blended version of metric and edge metric dimension.

Computational complexity and NP hardness of the metric dimension problem was discussed in [12, 13]. The metric dimension has a range of useful uses in our everyday lives, which motivates scholars to study it. The metric dimension is extensively used in the varied fields such as combinatorial optimization [14], pharmaceutical chemistry [15], robot navigation [16], sonar and coastguard loran, facility location problems, image processing [6] weighing problems [17], and computer networks [18]; for more practical applications, see the literature [19, 20].

The concept of metric dimension has a wide range of applications due to its versatility and also its generalizations which are edge, fault-tolerant, and mixed metric dimension are widely used to solve many difficult problems. The study of metric dimension on some cycle-based graphs are discussed in [21]. The preprint on the application of metric dimension and its algorithm is available at [22]. The metric dimension of chemical structures are discussed in [23]. The upper bounds for some chemical networks, one can find at [24]. The metric dimension and its related parameters are discussed in [25]. Generalized classes of some basic graphs are discussed in [26]. For the recent work on the computer 3D network ,we refer to [27]. For a detailed study on these parameters of metric dimensions, see [28]. The fault-tolerance of chemical topologies are discussed in [29]. The generalized resolving set for the circulant graph is available at [30, 31]. The famous convex structure’s resolving set can be seen at [32, 33]. The fault-tolerant of some computer networks are available at [34]. Some rotationally symmetric, extremal structures, and generalized Peterson graph are discussed in [35–37]. For the edge metric of polytopes structure, we refer to [38]. Th seminal works on edge and mixed metric are available at [10, 11]. All the resolvability parameters of graph theory on the polycyclic aromatic hydrocarbons are discussed in [39]. For the recent study of stepwise irregular graph, we refer to [40–43] and some important applications of irregular graphs and their measures [44–46]. For some topological work on this topic, one can find it in [47, 48].

Following are some important results which are very helpful to prove our main work.

Theorem 1. *Reference [11]: Letandbe the metric and fault-tolerant metric dimension of graph, respectively. Then,*

Theorem 2. *Reference [11]: Ifis a simple connected graph; then,.*

Theorem 3. *Reference [15]: Ifis a simple, undirected connected graph; then, (1) if .*

Theorem 4. *Reference [10]: For integers,, and. Moreover,ifis a path.*

Theorem 5. *Reference [11]: Ifis a graph with order; then,ifis a path.*

Figure 1 is the stepwise irregular unicyclic graphs having a minimum number of vertices and the maximum degree of a vertex is and due to equal order and size, its cyclomatic number is . It is the only graph having with even order and size. It is a unique graph in terms of even order of a stepwise irregular unicyclic graph [5]. To check the resolvability of the stepwise irregular graph, we consider this the minimum order and a special graph. The following lemmas start the discussion of the resolvability of the stepwise irregular graph.

Lemma 1. *If**is a unicyclic stepwise irregular graph with order 8, then*

*Proof. *To prove that , we will use the double inequality method. Let and the resolving set . In Table 1, we provide with regard to , the representations of whole vertex set are given as follows:

Since the representations in the above table are distinct and there are no two vertices with same representations, we can say that .

On the contrary, let . By Theorem.3, it is not possible, which concludes that the stepwise irregular graph in Figure 1 with

Lemma 2. *Let**be a unicyclic stepwise irregular graph with order 8. Then,*

*Proof. *On the contrary to prove that , first we will show that and the fault-tolerant resolving set . In Table 2, we provide with regard to , the representations of whole vertex set are given as follows:

Since, the representations in the above table are distinct and there are two vertices with same representations, we can say that .

On contrary to prove that we resulted to which is not possible by Theorem 1. It proves that the stepwise irregular graph that is defined in Figure 1 has

Lemma 3. *Let**be a unicyclic stepwise irregular graph with order 8. Then,*

*Proof. *On the contrary to prove that , let , and the edge resolving set . In Table 3, we provide with regard to , the representations of whole edge set are given as follows:

Since, the representations in the above table are distinct and we cannot see any two edges with same representations, we can say that .

Now, to prove the reverse inequality, on contrary and it is not possible by Theorem 1. It is concluded that the stepwise irregular graph in Figure 1 with

Lemma 4. *Let**be a unicyclic stepwise irregular graph with order 8. Then*

*Proof. *On the contrary to prove that , let , and the mixed metric resolving set . In Table 4, we provide with regard to , the representations of whole edge and vertex sets are given as follows:

Since, the representations in Table 4 are distinct and we cannot see any two components of the graph with same representations, we can say that .

Now, to prove reverse inequality , on contrary which is not possible by Theorem 5.

It is concluded that the stepwise irregular graph in Figure 1 with

The graph shown in Figure 2 is the stepwise irregular graph with order and size . Its cyclomatic number is . As shown in Figure 3, if we use , it create three cycles with order 4 and one cycle with order and it adds up to -cycles in a graph. In the following sections, we discuss different resolvability parameters of the stepwise irregular graph such as resolving of vertices (metric dimension) or edges (edge metric dimension) and also resolving of vertices along edges (mixed metric dimension). Moreover, the updated version of the resolving set, which is a fault-tolerant resolving set, is also discussed.

#### 2. Metric Dimension of a Stepwise Irregular Graph

This section refers to the resolving of vertices and computation of metric dimension of stepwise irregular graph .

Theorem 6. *Let**be a stepwise irregular graph with**. Then,*

*Proof 5. *To prove that when , we assume a resolving set . On the vertex set of , we build the following examples:From this representation, we obtain , that shows all the vertices of have unique representations with respect to resolving set .

For reverse inequality which is , by contradiction, it becomes . Following are some discussion for this claim.

*Case 1. *Consider the resolving set with where , then the same representations are either in or .

*Case 2. *Consider the resolving set where and , then the same representations are either in or .

*Case 3. *By assuming the resolving set where and , then the same representations are either in or or .

All discussion resulting in same representations of vertices. Hence, for ,

Theorem 7. *Let**be a stepwise irregular graph with**. Then,*

*Proof. *Assume that the resolving set . To show that , the induction will be applied to the number of copies of the graph. Assume that the assumption is valid for and that the base case is , as demonstrated in Theorem 6.We will prove that it is true for . We assume the following:Then, by using (13) and (15) in (16), we will obtain the following:Hence, it is true for all positive values of

#### 3. Fault-Tolerant Metric Dimension of

This section refers to the resolving of vertices with possibility of choosing one vertex extra for the replacement of faulty vertex in resolving set and also the computation of fault-tolerant metric dimension of stepwise irregular graph .

Theorem 7. *Let**be a stepwise irregular graph with**. Then,*

*Proof. *To prove that when , we assume a fault-tolerant resolving set . We construct the following cases on vertex set of As a result of the preceding reasoning in the form of representations that , all the vertices of have unique representations with respect to fault-tolerant resolving set .

For , our claim becomes contradictory as a result of the conflict which is not possible by Theorem 1. Hence, for ,

Theorem 8. *Let**be a stepwise irregular graph with**. Then,*

*Proof. *The fault-tolerant resolving set . Now to show that , the induction will be applied to the number of copies of the graph. We assume that the assumption is valid for and that the base case is , as demonstrated in Theorem 7.We will show that it is true for . SupposeUsing (19) and (22) in (23), we obtain the following:Hence, it is true for all positive values of

#### 4. Edge Metric Dimension of

This section refers to the resolving of edges of stepwise irregular graph ; it also for the computation of edge metric dimension of that graph.

Theorem 9. *If**be a stepwise irregular graph of order**, then**for .*

*Proof. *To prove that when , we assume an edge resolving set . We can see edge representations in Figure 2 and it has a distinct representation with respect to edge metric resolving set .

Hence, it follows that because all the edges of have unique representations with respect to edge resolving set .

For , our claim becomes contradictory as a result of the conflict . Following are some cases for this claim.

*Case 4. *Consider the edge resolving set with 2-cardinality where , then the same representations are either in or .

*Case 5. *Consider the edge resolving set again with 2-cardinality where and , then the same representations are either in or . All the cases resulted in contradiction, so this concludes that when ,

Theorem 10. *If**is a stepwise irregular graph of order**, then**for .*

*Proof. *The vertex and edge resolving set have similar vertices and same cardinality so that . To show that , the induction will be applied to the number of copies of the graph. Assuming that the assumption is valid for and that the base case is , as demonstrated in Theorem 9,We will show that it is true for . SupposeThen, by using (26) and (28) in (29), we obtain the following:Hence, it is true for all positive values of .

#### 5. Mixed Metric Dimension of

The mixed version of vertex and edge resolving is the mixed metric resolvability parameter. This section deals with resolving vertices as well as edges at a time.

Theorem 11. *Let**be a stepwise irregular graph with**. Then,*

*Proof. *To prove that when , we assume a mixed metric resolving set . We can see vertex representations from Theorem 6 and edge representations from Figure 3 which are showing that every vertex and edge has a distinct representation with respect to mixed metric resolving set .

Hence, it follows that , because all the vertices and edges of have unique representations with respect to mixed metric resolving set .

For , our claim becomes contradictory as a result of the conflict which is not possible by Theorem 2. So, all discussion concludes that when ,

Theorem 12. *Let**be a stepwise irregular graph with**. Then,*

*Proof. *The mixed metric resolving set is same as the vertex resolving set and edge resolving set with same cardinality so that . To show that , the induction will be applied to the number of copies of the graph. We assume that the assumption is valid for and that the base case is , as demonstrated in Theorem 11.We will show that it is true for . SupposeUsing (32) and (34) in (35), we obtainHence, it is true for all positive values of .

Figure 2 is the stepwise irregular graph with , the labeling of vertices can be extended similarly to define in the figure and can be any natural number greater than 2. Moreover, the dark black vertices show the vertex, edge, and mixed metric resolving set when and the position of resolving set vertices remains the same while increasing the number of copies of ’s.

#### 6. Conclusion

Recently, the generalized class of graphs known as stepwise irregular graphs was introduced. In this research, we initiate to discuss the resolvability parameters of this new type of graphs. One of the generalized stepwise irregular graphs is as shown in Figure 3. Each resolving parameter of is not constant, the cardinality of each resolving parameter increases with increasing graph’s cyclomatic number which is . As we have seen that the resolving parameters are also dependent on the degree of vertices of a graph and the stepwise irregular class of graphs is purely dependent on the degree of vertices, so the discussion of resolvability parameters of stepwise irregular graphs is interesting and opens the way of thinking on either stepwise irregular graphs or resolvability parameters of graphs. To extend this idea, we provide the following open problems at the end of this work. [50, 51].

*Open problem 1. **Do all the stepwise irregular graphs have varying resolving parameters**?*

*Open problem 2. **Is there any class of stepwise irregular graph with constant resolving parameters(**?*

#### Data Availability

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#### Conflicts of Interest

The authors declare that they have no conflicts of interest.