On the Inverse Problem for Some Topological Indices

Maji, Durbar; Ghorai, Ganesh; Mahmood, Muhammad Khalid; Alam, Md. Ashraful

doi:https://doi.org/10.1155/2021/9411696

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Volume 2021 | Article ID 9411696 | https://doi.org/10.1155/2021/9411696

On the Inverse Problem for Some Topological Indices

Durbar Maji,¹Ganesh Ghorai,¹Muhammad Khalid Mahmood,²and Md. Ashraful Alam³

Academic Editor: Lazim Abdullah

Received11 Sept 2021

Revised20 Oct 2021

Accepted22 Oct 2021

Published30 Nov 2021

Abstract

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value to every graph within a given family of graphs, find some for which . It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated -index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.

1. Introduction

Throughout the paper, we consider as a simple (without loops and multiple edge loops) finite graph that contains vertices and edges, respectively. The notation denotes the degree of a vertex . All other notations and terminologies used but not clearly stated in this paper may be followed from [1].

In chemical graph theory, a TI, usually known as a molecular descriptor, can be expressed by a real number calculated from a chemical/molecular graph which is the representation of a chemical compound by replacing atoms with vertices and bonds with edges. The TI is calculated for evaluating the information about the atomic constitution and bond characteristics of a molecule/chemical compound. The TI of a molecular graph is a numerical number that enables us to collect information about the concerned chemical structure. It helps us to know its hidden properties without performing experiments [2–4]. The TIs also correlate and predict several physical, chemical, biological, pharmaceutical, pharmacological activities/properties from molecular structures of graphs corresponding to real-life situations. The IP is defined as the feasibility of finding/modeling the chemical structure represented by a graph whose index value is equal to a given non-negative integer for the integer-valued problem. In the QSAR and QSPR studies [5], a method by which it is possible to predict the properties of a given molecular structure is called a forward problem. The inverse problem is concerned that, one can design the exact molecular structure that satisfies the given target properties by applying the forward problem solution.

The most popular as well as the oldest degree-based graph indices are the first and second Zagreb indices. Gutman et al. introduced the first Zagreb index in [6] and second Zagreb index in [7]. They are defined, respectively, as

In 2016, Farahani et al. [8] defined the second hyper-Zagreb index as follows:

Kulli [9] introduced the first and second Gourava indices, defined, respectively, as

Milicevic proposed the reformulated first Zagreb index in [10], defined as

Liu et al. [11] put forward the reformulated -index, and it is defined as follows:

Alameri et al. [12] introduced -index, and it is defined by

Graph theory, a branch of mathematics, provides the tools for solving problems of information theory, computer sciences, physics, and chemistry [13–15]. The study of the IP is encountered in various fields of science, especially in mathematics and chemistry. The IP for a TI is defined as follows: for a given TI and a non-negative integer value , find (chemical) graph for which . Also, the inverse existence problem [16] for the pair can be asked as follows: given a class of graph and function , for which there is with ? The idea of the IP based on TIs was initiated by the Zefirov group in Moscow [5, 17, 18] and was first proposed by Gutman et al. in [19]. The IP for Wiener index was solved in [20]. In [21], the IP was studied for sigma index as well as for acyclic, unicyclic, and bicyclic graphs. The same problem for the Steiner Wiener index was also solved in [22]. This type of problem for the Zagreb indices, forgotten Zagreb index, and the hyper-Zagreb index was studied in [23]. Tavakoli et al. [24] addressed the IP for first Zagreb index. Also, the IP for some graph indices was investigated in [25]. In [26], Czabarka et al. solved the IP for certain tree parameters. To study more about the inverse problems and the topological indices of graph operations, one can see these references [27–32].

There are so many benefits in the solution of IP. It helps to design of combinatorial libraries for drug discovery in combinatorial chemistry [33]. It may be helpful in speeding up the discovery of lead compounds with desired properties [34]. Also, the IP plays its significance in the application of trees [35] such as the field of algorithms, chemical graph theory, signal processing, and electrical circuits.

2. Preliminaries

To study the IP for -index, we will use the following crucial observation.

Let Ł be a graph having and as vertices which are adjacent to each other. We subdivide each edge by introducing a new vertex (of degree 2) to construct a new graph Ł^∗ (see Figure 1).

Here, the -index of the new graph Ł^∗ will be sixteen which is more than the graph Ł.

Lemma 1. By applying the transformation , the -index value will be increased by 16. That is, .

Proof. Let us consider the graph Ł with vertices and of degrees and , respectively. Since in the new constructed graph Ł^∗, a new vertex is inserted between and ,

Lemma 2. If either or (or both), then by the means of the above transformation , the value of the first Gourava index increases by 8. That is,Similarly, the value of the second Gourava index increases by 16. Thus,

Lemma 3. If either or (or both), then by means of the above transformation , the value of the second hyper-Zagreb index increases by 16. That is,

Lemma 4. If either or (or both), then by applying the transformation , the values of the reformulated first Zagreb index and reformulated -index of the graph Ł increase by 4 and 8, respectively. Thus,

3. Main Results and Discussion

In graph theory, the IP based on the TIs is an interesting one among the problems associated with the estimation of different graph invariants/TIs such as chromatic number, connectivity, girth, and number of independent sets. The IP plays a crucial role in many areas of science, especially in mathematics. In this section, we have investigated the IP for -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated -index. Additionally, we have developed the same problem for molecular, tree, acyclic, unicyclic, and bicyclic graphs.

3.1. The IP for -Index

Here we will discuss the IP for -index.

Theorem 1. The -index for connected graphs can take all positive even integers, except for where ; and .

Proof. Proof. To prove the theorem, we establish a set of graphs whose -index values are 48, 2, 84, 166, 248, 330, 412, and 494, respectively. These numbers are congruent to 0, 2, 4, 6, 8, 10, 12, and 14 (mod 16), respectively.
Consider the cyclic graphs for . Clearly, in Figure 2(a), , , , and for . Now we apply Lemma 1 for each graph in Figure 2. Thus, takes all those even positive integer values which are divisible by 16, except 16 and 32.
In Figure 2(b), . By applying the transformation in Lemma 1, we arrive at graphs whose -index values are , and so on. There are the path graphs. Thus, takes all positive even integer values which are congruent to 2 (mod 16).
Now consider the graph in Figure 2(c) with . By applying Lemma 1, we can obtain graphs with -index values . Here, contains all positive even integer values 4 (mod 16), except the integers .
The graph is depicted in Figure 2(d) with , and then by using Lemma 1, we can take the graphs with -index values , and so on. Therefore, covers all positive even integer values 6 (mod 16) and . Obviously, the integers are not covered by the construction.
In Figure 2(e), we have , and then by using Lemma 1, we can take the graphs with -index values , and so on. So, takes all positive even integer values 8 (mod 16) and . Obviously, the integers are not covered by the construction.
The graph in Figure 2(f) contains . Again by Lemma 1, goes to all those positive even integer values 10 (mod 16) and giving 330, 348, 364, 380, …, and so on.
From Figure 2(g), we have . By applying Lemma 1, takes all positive even integer values 12 (mod 16) and also having 412, 428, 444, 460, …, and so on.
The graph in Figure 2(h) has . By applying Lemma 1, takes all positive even integer values 14 (mod 16) and also taking 494, 510, 526, 542, …, and so on. There exist no connected graphs with the -indices mentioned in Table 1.

Corollary 1. The -index of a tree (or molecular) graph can take all positive even integers, except for where ; and .

Corollary 2. Let Ł be a connected unicyclic or bicyclic graph. Then, there exists the -index of the form for all non-negative integers where and .

Proof. Proof. Let be a unicyclic graph which is obtained by adding a path of length two to any vertex of the cyclic graph for . Thus, . Similarly, the unicyclic graph is obtained by adding another path of length two to one of the two adjacent vertices to that lie on (see Figure 3). We get .
The bicyclic graph is obtained by gluing two cyclic graphs with one side of both, and we have which also can be expressed in the said form. Similarly, another bicyclic graph is obtained by adding a path with length two to any vertex of . Thus, is of the form for .

(a)

(b)

(c)

(d)

3.2. The IP for Gourava Indices

Here we study the IP for Gourava indices.

Theorem 2. The first Gourava index of a connected graph can take any positive integer except for 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 27, 28, 29, 31, 33, 35, 41, 44, 49, 57, and 73.

Proof. By applying Lemma 2, we construct eight series of graphs containing at least one vertex of degree 2, whose -values are of the form for . These graphs are drawn in Figures 4(a)–4(h).
Consider a cycle graph with . Clearly, . Therefore, can take all those positive integer values which are divisible by 8. From the graph (Figure 4) and Lemma 2, we obtain the graphs with -values . Thus, takes all positive integers (mod 8) and .
Consider the graph in Figure 4 with , i.e., implies all positive integer values (mod 8). Then, by Lemma 2, we obtain graphs with = 89, 97, 105, 113, 121, …. Again for the graph , . Thus, we can arrive at graphs whose values are 18, 26, 34, 42, 50, 58, 66, 74, 82, …. Similarly, by applying Lemma 2 to the graphs , , , , and , we get the graphs with = 59, 67, 75, 83, 91, 99, 107, 115, …, 60, 68, 76, 84, 92, 100, 108, …, 45, 53, 61, 69, 77, 85, …, 38, 46, 54, 62, 70, 78, …, and 47, 55, 63, 71, 79, 87, …, respectively. The star graphs , and examples depicted in Figures 2(d) and 5(b) show that there exist graphs with as 21, 36, 43, and 65, respectively. There exist no connected graphs with first Gourava index as listed in Table 2.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(a)

(b)

Corollary 3. The first Gourava index of a tree (or molecular) graph can take any positive integer, except for 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 27, 28, 29, 31, 33, 35, 41, 44, 49, 57, and 73.

Now we study to settle the IP for the second Gourava index.

Theorem 3. The second Gourava index of a connected graph can take any positive even integer except for 4, 6, 8, 10, 14, 16, 20, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 52, 54, 56, 58, 62, 66, 68, 70, 72, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 118, 122, and 134.

Proof. Consider first path graphs for . It is clear that and . By Lemma 2, since for , we obtain the graphs whose -values are 28, 44, 60, 76, 92, 108, … which are congruent to 12 (mod 16). If we consider a cyclic graph with vertices, then for . Clearly, we get the graphs with -values 48, 64, 80, 96, 112, …. Now the graph drawn in Figure 4 has . Therefore, by applying Lemma 2, we obtain the graphs with second Gourava index values 100, 116, 132, 148, 164, …. The graphs and in Figure 4 contain and , respectively. Therefore, by Lemma 2, obtains all those even integer values 126, 142, 158, 174, 190, 206, 222, … and 104, 120, 136, 152, …, respectively. In same procedure, from Figure 4, we get the graph with . So, it follows the -values 154, 170, 186, 202, 218, ….
In Figure 6, by Lemma 2, we have GO₂ () = 150 and GO₂ () = 114 with. Then by using the Lemma 2, we can take the graphs with 166, 182, 198, 214, … and 130, 146, 162, 178, 194, …, respectively. The integers not covered by the above transformation for the second Gourava index are listed in Table 3.

Corollary 4. The second Gourava index of a tree (or molecular) graph can take any positive even integer, except for 4, 6, 8, 10, 14, 16, 20, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 52, 54, 56, 58, 62, 66, 68, 70, 72, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 118, 122, and 134.

3.3. The IP for Second Hyper-Zagreb Index

Theorem 4. The second hyper-Zagreb index of a connected graph can be any positive integer, except 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 101, 102, 103, 107, 109, 110, 111, 114, 115, 118, 119, 123, 126, 127, 130, 131, 133, 134, 135, 141, 142, 143, 146, 147, 149, 150, 151, 157, 158, 159, 163, 165, 167, 173, 174, 175, 181, 183, 190, 191, 199, 206, 215, 222, 223, 231, 239, 255, and 271.

Proof. We construct a series of sixteen graphs such as . The values of these graphs can be expressed in the form for . In Figure 7, we construct the graphs for and . Note that the graphs are similar to Figures 4(a), 4(f), 4(c), 4(h), and 4(g) respectively. We have () = 48. Then by using Lemma 3, we can take the graphs with values 64, 80, 96, 112, … which are congruent to 0 (mod 16). Secondly, consider the graph which generates the -values , and all these are congruent to 1 (mod 16).
Since () = 162, by mean of the construction method upon the graph H₂ from the Lemma 3, we get the graphs with HM₂ values 178, 194, 210,226, … which are congruent to 2 (mod 16). For , the -values cover the integers which are (mod 16). Analogously, the values for the graph cover all the integers ., and all these numbers are congruent to 4 (mod 16). For , it provides the values which are (mod 16). For there exist values which are (mod 16). It follows for the graph , and these integers are congruent to 7 (mod 16). Similarly by Lemma 3, we get for the graphs with Now, using the construction method in Lemma 3 to , and , we can generate the graphs with values 8, 24, 40, … congruent to 8 (mod 16); 89, 105, 121, … congruent to 9 (mod 16); 58, 74, 90, … congruent to 10 (mod 16); 139, 155, 171, … congruent to 11 (mod 16); 108, 124, 140, … congruent to 12 (mod 16); 189, 205, 221, … congruent to 13 (mod 16); 238, 254, 270, … congruent to 14 (mod 16); and 257, 303, 319, … congruent to 15 (mod 16), respectively.
The examples drawn in the Figures 2(c), 2(d), 5(a), and 5(b) have values , and 207, respectively. These numbers are congruent to , and 15 (mod 16), chronologically. There exist no connected graphs with the -indices mentioned in Table 4.

Corollary 5. The second hyper-Zagreb index of a tree (or molecular) graph can be any positive integer, except 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 101, 102, 103, 107, 109, 110, 111, 114, 115, 118, 119, 123, 126, 127, 130, 131, 133, 134, 135, 141, 142, 143, 146, 147, 149, 150, 151, 157, 158, 159, 163, 165, 167, 173, 174, 175, 181, 183, 190, 191, 199, 206, 215, 222, 223, 231, 239, 255, and 271.

3.4. The IP for Reformulated First Zagreb Index

Theorem 5. The first reformulated Zagreb index of a connected graph can take all positive even integer values except for 4 and 8.

Proof. At first, we consider the path with . The -value for is equal to 2. By Lemma 4, we obtain graphs with -values 6, 10, 14, 18, …, and so on. Also, since , by means of the construction described in Lemma 4, we have the graphs whose -values are 16, 20, 24, 28, 32, 36, 40, …. Hence, covers all positive even integers except 4 and 8.

Corollary 6. The first reformulated Zagreb index of a tree (or molecular) graph can take all positive even integer values, except for 4 and 8.

3.5. The IP for Reformulated -Index

Theorem 6. The reformulated -index of a connected graph can be any positive even integer, except for 4, 6, 8, 12, 14, 16, 20, 22, 28, 30, 36, 38, 46, 54, and 62.

Proof. For the path graphs , we get . Thus, by applying Lemma 4, we obtain graphs with -values equal to 10, 18, 26, 34, …. Similarly, for the cyclic graph , we have and next obtained graphs having -values 32, 40, 48, 56, …. In an analogous manner, starting with and in Figure 4, we obtain the graphs with -values 78, 86, 94, 102, … and 52, 60, 68, 76, …, respectively.

Corollary 7. The reformulated -index of a tree (or molecular) graph can be any positive even integer, except for 4, 6, 8, 12, 14, 16, 20, 22, 28, 30, 36, 38, 46, 54, and 62.

4. Conclusion

The inverse problem is one of the recent problems of graph theory related to the applicative area. Here, we have studied the IP based on some topological graph indices such as -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated -index. We have studied the inverse problems for the aforesaid indices since they are closely related to each other. We have also investigated the results for tree, molecular, unicyclic, and bicyclic graphs. The inverse problem is still open for other graph indices and other molecular structures.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

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