Abstract
Let be a graph. Denote by , the degree of a vertex of and represent by , the edge of with the end-vertices and . The sum of the quantities over all edges of is known as the symmetric division deg (SDD) index of . A connected graph with vertices and edges is known as a (connected)-cyclic graph. One of the results proved in this study is that the graph possessing the largest SDD index over the class of all connected-cyclic graphs of a fixed order must have the maximum degree . By utilizing this result, the graphs attaining the largest SDD index over the aforementioned class of graphs are determined for every . Although, the deduced results, for , are already known, however, they are proved here in a shorter and an alternative way. Also, the deduced results, for , are new, and they provide answers to two open questions posed in the literature.
1. Introduction
Only finite and connected graphs are discussed in this study. The graph-theory concepts used in this paper without defining them here can be found in [1, 2]. A graph invariant is a mapping on the class of all graphs with the condition that whenever the graphs and are isomorphic to each other. In chemical graph theory, graph invariants are usually referred to as topological indices.
The present paper is concerned with a topological index devised in [3], under the name “symmetric division deg (SDD) index,” for improving QSAR/QSPR modeling. The SDD index of a graph is defined aswhere denotes the edge set of , indicates the edge with end-vertices , and denotes the degree of an arbitrary vertex . In order to check the chemical applicability of the SDD index, Furtula et al. [4] conducted a thorough multidimensional investigation of the SDD index and discovered that it is a practicable and feasible topological index that outperforms a number of other indices of a similar kind. Ali et al. and Sun et al. [5, 6] provide information on certain extremal results concerning the SDD index. The papers [7,15] may be consulted for different bounds on the SDD index.
One of the results proved in this paper is that the graph possessing the largest SDD index over the class of all -cyclic graphs of a fixed order must have the maximum degree . By utilizing this result, the graphs attaining the largest SDD index over the aforementioned class of graphs are determined for every . Although, the deduced results, for , are already known [8, 9], however, they are proved here in a shorter and an alternative way. Also, the deduced results, for , are new, and (the case gives a solution to an open problem given in [10]) they provide an answer to a conjecture posed by Palacios [11].
2. Main Results
We start this section by defining a graph transformation that is used in proving one of the main results of this paper.
Transformation 1: let and be two vertices of a graph such that (i) , for every , and (ii) has at least one neighbor that is not adjacent to . Take . Form a new graph from by inserting the edges and dropping the edges .
We also need the following elementary lemma.
Lemma 1. DefineThe function is a strictly increasing in , but it is strictly decreasing in .
Proof. Take two numbers and satisfying . Note that , which gives , and thus, . In a similar way, one concludes that whenever .
Proposition 1. For , the maximum degree of is if it is a graph possessing the largest SDD index among all -cyclic graphs of order .
Proof. Contrarily, we consider the case when the maximum degree of is different from . Choose in such a way that , for every . Certainly, there exist so that the set contains at least one element, different from v, that is not a neighbor of . Suppose that . Let be the graph obtained by applying Transformation 1 on . Take and . In what follows, for an arbitrary vertex , by the vertex degree , we mean the degree of the vertex in :where Lemma 1 contains the definition of . Now, keeping the facts into consideration that and , one obtainsNote that and , for every and , respectively. Thus, because of the above observations and Lemma 1, Equation (3) yields ; this contradicts the definition of because is also a -cyclic graph of order .
Proposition 1 confirms that the graph(s) possessing the largest SDD index among all -cyclic graphs of order must be obtained from the star graph by adding edge(s) in some way. Since there is exactly one -cyclic graph of order with the maximum degree for every and , the next already known result follows immediately from Proposition 1.
Corollary 1 (see [8, 9]). The graph uniquely attains the largest SDD index among all -cyclic graphs of order for every and , where is the graph formed by inserting edge in the star graph .
Note that there are exactly two bicyclic graphs of order with the maximum degree for every ; one of them, say , is formed by adding two adjacent edges in the star graph , and the other one, say , is formed by inserting two nonadjacent edges in . Sincefor every , the next already known result follows also immediately from Proposition 1.
Corollary 2 (see [9]). The graph uniquely achieves the largest SDD index over the family of all bicyclic graphs of order for every .
We note that there is only one tricyclic graph, namely, , of order 4. For , one obtains all the possible tricyclic graphs of order and maximum degree from Figure 1. Simple computations yieldWe observe that , for . However,
Therefore, we have another corollary of Proposition 1.
Corollary 3. The graph , shown in Figure 1, attains uniquely the largest SDD index over the family of all tricyclic graphs of order , for every .
We remark here that Corollary 3 gives a solution to an open problem posed in [10]. Next, we discuss the tetracyclic graphs. For , Figure 2 contains all the possible tetracyclic graphs of order and maximum degree . After elementary computations, we haveWe observe that , for , and
Thus, the next result yet is another consequence of Proposition 1.
Corollary 4. The graph (, respectively) pictured in Figure 2, attains uniquely the largest SDD index over the family of all tetracyclic graphs of order , for every (for every , respectively).
Finally, we discuss the pentacyclic graphs. We note that there is only one pentacyclic graph of order 5, namely, depicted in Figure 2. For , Figure 3 contains all the possible pentacyclic graphs of order and maximum degree . After elementary computations, we haveWe observe that , for . Also, it holds thatfor . Moreover, , for , and , for (the graph does not exist when ). Thus, by keeping in mind these observations and using Proposition 1, one gets the next result.
Corollary 5. The graph (, respectively) given in Figure 3, attains uniquely the largest SDD index over the family of all pentacyclic graphs of order for every (for every , respectively).
3. Concluding Remarks
In this study, we have proved that the graph possessing the largest SDD index over the class of all -cyclic graphs of a fixed order must have the maximum degree . By utilizing this result, we have determined the graphs possessing the maximum SDD index over the aforementioned class of graphs for . Although, the obtained results, for , are already known [8, 9], however, they are proved here in a shorter and an alternative way. Also, the obtained results, for , are new, and (the case gives a solution to an open problem given in [10]) they provide an answer to the following conjecture posed by Palacios [11].
Conjecture 1. For and sufficiently large , among all -cyclic graphs of a fixed order , the greatest values of the SDD index and the graph invariant are achieved by the same graph .
Here, we remark that the graph invariant is a special case of the zeroth-order general Randić index (for example, see [12]), where is a real number distinct from 1 and 0. From Corollaries 3–5, and from the results obtained in [13, 14] involving the maximum value of graph invariant , we deduce that Conjecture 1 is true for (i) and , (ii) and , and (iii) and . It is natural to expect that Conjecture 1 holds also not only for but also for ; however, the proof of this expected result is left here is an open task.
Data Availability
The data used to support the findings of the study can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors declare that they do not have any conflicts of interest regarding the publication of this paper.