Abstract
Topological index (TI) is a mapping that associates a real number to the under study (molecular) graph which predicts its various physical and chemical properties. The generalized degree distance index is the latest developed TI having compatible significance among the list of distance-based TIs. In this paper, the minimum generalized degree distance of unicyclic, bicyclic, and four cyclic graphs is determined. Mainly, the associated extremal (minimal) graphs are also identified among all the aforesaid classes of graphs.
1. Introduction
Let class of n-vertices connected graphs is denoted by . Then, represents the subclass of with linearly independent cycles and edges. In this paper, is considered. For any graph , represents the shortest distance between the vertices , and the maximum of for any is defined to be the diameter of , denoted by . A well-known topological index is the Wiener index, which gives the sum of distances between all pairs of vertices of a graph. A new graph invariant named degree distance was introduced by Dobrynin and Kotchetova [1] and Gutman [2] and defined as
For a graph , an additively weighted Harary index is given by [3]
For every vertex a, the generalized degree index denoted by is defined as follows:where . For a graph ,where is a real number. Let be a family of graphs a graph is called extremal graph if or .
1.1. Research Gaps and Motivation
Asma et al. found the minimum generalized degree distance of tricyclic graphs in [5]. Moreover, Jianzhong et al. [6] have found degree distance topological indices for derived graphs. One can also find the results on the degree distance of strong products of graphs in [7]. This suggests that there is still room for the research on the topic of the minimum generalized degree distance of -cyclic graphs for , and 4.
1.2. Novelty and Contributions
In this paper, all the extremal unicyclic, bicyclic, and four cyclic graphs having minimum generalized degree distance are determined. Throughout this paper, and denote the class of unicyclic, bicyclic, and four cyclic graphs on n vertices, respectively.
2. Applications
The topological indices find their application in the areas of chemistry such as drug discovery, finding the physio-chemical properties of compounds such as melting point, boiling point, and -electron energy. Also, they are helpful in providing the correlation between the aforesaid properties of chemical compound and thermodynamical properties. Moreover, it explains the molecular branching and cyclicity of chemical compound. Moreover, it also establishes correlations with various parameters of chemical compounds. To find more on their applications in chemical strata, see [4, 8, 9].
3. Classification of Cyclic Graphs
The characterizations of connected unicyclic, bicyclic, and 4-cyclic graphs by their degree sequence are given as follows.
Lemma 1 (see [10]). The degrees of vertices of a unicyclic graph are the integers , if and only if:(i)(ii)(iii), for at least three indices
Lemma 2 (see [10]). The degrees of the vertices of a bicyclic graph are the integers if and only if:(i)(ii)(iii), for at least four indices(iv).
Lemma 3 (see [11]). The degrees of the vertices of a four cyclic graph are the integers if and only if:(i)(ii)(iii), for at least five indices.
Let the number of vertices of graph of degree is denoted by , for . If , then
Let us denote
To determine the minimum of over all integers , which satisfy the conditions of above three lemmas.
Thus, Lemma 1-Lemma 3 with the help of aforesaid notions can be rewritten as follows:
Lemma 4 (see [10]). The integers are the multiplicities of the degrees of a unicyclic graph if and only if:(i)(ii)(iii)(iv)
Lemma 5 (see [10]). For bicyclic graph, the integers represent the multiplicities of the degrees of vertices if and only if:(i)(ii)(iii)(iv)
Lemma 6 (see [11]). The integers are the multiplicities of the degrees of a four cyclic graph if and only if:(i)(ii)(iii)(iv)The set of vectors , which satisfy the conditions of Lemma 4, 5, and 6, is denoted by , and , respectively.
Now, we consider the transformation , which is defined for , , , , , as follows [10]: .
We have for .
Let and the transformation defined as for for .
Lemma 7. Suppose is a positive integer and , for .(a)(b)
Proof. (a)As and . If , for and and , then a contradiction. Also, if , and and , then a contradiction.(b)By simple calculations, .
Lemma 8. Suppose and .(a)(b)
Proof. (a)Proof of (a) is the same as above(b)By putting in the above, it holds that
4. Main Result
This section deals with the main results related to our finding of the minimum generalized degree distance index for the different families of the cyclic graphs.
Theorem 9. For every and , it holds thatand the unique extremal graphs is .
Proof. For , the only unicyclic graph is and .
For , if , we will get at least two cycles that do not satisfy the hypothesis. Thus, . Next, we investigate the values of for . If and , then by applying the transformation at position and , we get a smaller value of . Now, for , the value of . If , then which is not possible. Since , first we consider , then and imply that and which corresponds to the graph . If , then the conditions of Lemma 4 imply that and , and hence,and the unique extremal graphs is .
Theorem 10 (see [10]). For every and , it holds that
Proof. By putting in Theorem 9, the above result is proved, and the result is the same as Theorem 3.1 in [10].
Theorem 11. For every and , it holds thatand the unique extremal graphs are obtained from by adding two edges of common extremity.
Proof. For , the unique bicyclic graph is with an edge and .
For , it holds that we have and . Since , first we consider , then and imply that . If , then by action of transformation at position 3, a smaller value for is determined. Consider if , then and . If , we have and . If , then we get , and hence,and the unique extremal graphs is with two edges of the common vertex.
Theorem 12 (see [10]). For every and , it holds that
Proof. By putting in Theorem 11, the above result is proved, and the result is the same as Theorem 3.2 in [10].
Theorem 13. For and , it holds that
Then, all the extremal graphs are isomorphic to . The graph is obtained by identifying the center of star with an arbitrary vertex of degree 5.
Proof. In order to find , it is enough to find , where . Let , only graphs (see Figure 1). Also, and . Let us consider . For , all graphs are where shown in Figure 2. For these graphs which hold for the graph .
Finally, for . If , Then, we have at least five cycles; hence, must be less than or equal to one.
Now, we investigate the possible values of . If there exists , such that and , then by the action of at position and , a new vector for which is obtained.
Similarly, if there exists such that , a new degree sequence in is determined by which . Now, we consider two cases:
Case 14. Consider distinct indices and such that and . If , since , we will analyze the two cases separately.(a)In this case, , where and . By considering different vertices in such a way that . The vertices are adjacent to and . Also, and are adjacent. Then, there exist five cycles which contradicts the hypothesis.(b)Suppose , then and and is characterized by the equations and , which implies that , by solving for and , and then by applying the transformation for position 2 or 3, we obtain a smaller value of .
Case 15. Suppose that holds, the degree sequence is . As , so we have to analyze two cases:(a)If , then . This equation does not hold. If all , and are not greater than 2, then , which contradicts the hypothesis . If for any by applying at position , the minimum value of is obtained.(b)If , then . If , then . So , if , then , which implies that , and which is a contradiction as . So, . Thus, either or .If , then , the only possible solution that follows Lemma 6 and gives a graphical degree sequence is . Thus, andNext, consider if , then . There are only two possible solutions that satisfy the conditions of four cyclic graph. These graphical sequences are and . By applying at position 3 of , we obtain a degree sequence andSince Hence,and the unique extremal graph is obtained by identifying the center of graph with an arbitrary degree 4 vertex of graph (Figure 2).
Theorem 16. Let , then
Proof. By putting in Theorem 13, the above result is proved, and the result is the same as Theorem 12 in [11].
5. Conclusion
In this note, we have computed the minimum generalized degree distance indices in the different families of unicyclic, bicyclic, and four cyclic graphs. The extremal graphs having minimum generalized degree distance indices are also characterized among these families of graphs. However, the problem is still open to compute this index for various families of -cyclic graphs for and [12–17].
Data Availability
The data supporting the current study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.