Abstract
It is considered that there is a fascinating issue in theoretical chemistry to predict the physicochemical and structural properties of the chemical compounds in the molecular graphs. These properties of chemical compounds (boiling points, melting points, molar refraction, acentric factor, octanol-water partition coefficient, and motor octane number) are modeled by topological indices which are more applicable and well-used graph-theoretic tools for the studies of quantitative structure-property relationships (QSPRs) and quantitative structure-activity relationships (QSARs) in the subject of cheminformatics. The -electron energy of a molecular graph was calculated by adding squares of degrees (valencies) of its vertices (nodes). This computational result, afterwards, was named the first Zagreb index, and in the field of molecular graph theory, it turned out to be a well-swotted topological index. In 2011, Vukicevic introduced the variable sum exdeg index which is famous for predicting the octanol-water partition coefficient of certain chemical compounds such as octane isomers, polyaromatic hydrocarbons (PAH), polychlorobiphenyls (PCB), and phenethylamines (Phenet). In this paper, we characterized the conjugated trees and conjugated unicyclic graphs for variable sum exdeg index in different intervals of real numbers. We also investigated the maximum value of SEIa for bicyclic graphs depending on .
1. Introduction
In chemical graph theory, molecules and macromolecules (such as organic compounds, nucleic acids, and proteins) are represented by graphs wherein vertices correspond to the atoms, whereas edges represent the bonds between atoms [1, 2]. A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physicochemical properties [3]. Topological indices play a significant role in organic chemistry and particularly in pharmacology [4, 5]. Physicochemical properties of chemical compounds such as relative enthalpy of formation, biological activity, boiling points, melting points, molar refraction, acentric factor, octanol-water partition coefficient, and motor octane number are modeled by topological indices in quantitative structure-property relation (QSPR) and quantitative structure-activity relation (QSAR) studies [4, 6–8].
In chemistry, the usage of topological index started in 1947 when the chemist Wiener developed the Wiener index (a distance-based topological index) to predict boiling points of paraffins [9]. Platt index (the oldest degree-based topological index) was proposed in 1952 for predicting paraffin properties [10]. The -electron energy of a molecular graph was calculated by adding square of degrees (valencies) of its vertices (nodes) in the year 1972. The same computational result, afterwards, was named the first Zagreb index, and in the field of molecular graph theory, it turned out to be a well-swotted topological index [11]. For more details about the topological indices in the field of chemistry, we refer to [6, 8, 12–15].
Many well-known topological indices such as hyper Zagreb index [16], variable sum exdeg index [17], and Zagreb indices [18, 19] have been used to find out sharp bounds for unicyclic, bicyclic, and tricyclic graphs. Vukicevic [15] propounded variable sum exdeg index for a graph and defined it aswhere is a positive integer other than 1. This topological index is correlated well with octane-water partition coefficient [15] and is employed to the study of octane isomers (see [20–22]). This topological index in the form of polynomial was proposed by Yarahmadi and Ashrafi, and they find its application in nanoscience [23]. Chemical application of this index can be seen in the papers [12, 13, 15].
In this paper, we mainly targeted three main problems. First of all, we find the extremal values of variable sum exdeg index () for conjugated trees. After that, we investigated lower and upper bounds of unicyclic conjugated graphs with respect to the length of this cycle in different intervals. At the end of this paper, we find upper bounds of for bicyclic graphs. This paper contains seven sections. In the first section, we have given introduction while in Section 2, we have given the proofs of some lemmas and preliminary results. In Section 3, we discovered the bounds of a conjugated trees and this section helps us to find out lower and upper bounds of unicyclic conjugated graphs with respect to the length of this cycle in Section 4. In Section 5, we discussed an important theorem related with conjugated unicyclic graphs. In Section 6, we discovered the upper bounds of bicyclic graphs. In the last section, we have drawn the conclusion.
2. Preliminary Results
All graphs under consideration in this paper will be connected, simple, and finite. Suppose is a simple and finite graph, whereas set of vertices is denoted by and the set of edges is denoted by . Let for which is defined as the cardinality of edges incident with the vertex . Suppose denotes the set of all vertices which are adjacent with the vertex and . Note that and represent the maximum and minimum degree of a graph , respectively. A pendent vertex is a vertex of degree one. An edge whose one end is a pendent vertex is called pendent edge. Let and ; then, and are subgraphs of which are obtained by deleting the vertices and edges from, respectively. An edge between the vertices and is denoted by . If and , then and can be expressed as and , respectively.
In a graph , if the vertices and are nonadjacent, then means there is an addition of an edge between the vertices and in a graph . We use , and to denote the star graph, cycle graph, and path graph on vertices, respectively. We assume that graphs and be rooted at and , respectively. Then, is obtained by identifying and as the same vertex. A graph which has no cycle is called a tree. A graph is said to be unicyclic graph if it has a unique cycle. A graph is said to be bicyclic graph if has exactly edges. Let represent the collection of all those graphs which have order and a unique cycle of length . We denote the collection of all conjugated unicyclic graphs of order in which length of its cycle is , whereas is the matching number of . Let be a unicyclic graph of length and it is denoted by . Let ; if or , then its is unique. That is why in this paper we will assume . One can find terminologies and expressions “indefinito” in [24–26].
Suppose that is a graph acquired from another graph by using some graph alteration such that . In all sections of this paper, whenever such two graphs are under debate, we always mean the vertex degree the degree of the vertex in .
Lemma 1. Let be a graph of order if contains the vertices such that and ; then, there exists a graph such that for .
Proof. Let and be the pendent vertices adjacent to the vertex . We define a new graph , i.e., as in Figure 1. By the definition of , we havewhere for . Thus, the proof of the above lemma is accomplished.
(a)
(b)
Lemma 2. Let be a graph having two components and , where is a cycle graph and is a star graph with central vertex . Let and , such that is an edge in . Let be the pendent vertices adjacent with the vertex , i.e., . We define such that .
Proof. Let be a graph having two components and where is a cycle graph and is a star graph with central vertex . Let , , such that is an edge in . Let be the pendent vertices adjacent with the vertex , i.e., . We define as in Figure 2. By the definition of , we haveIf , thenwhere and .
If , thenwhere and . Thus, we have .
(a)
(b)
3. Extremal Values of Variable Sum Exdeg Index for Conjugated Trees
First we introduce some notations which will be used in the following lemmas and theorems. Suppose that be the collection of all trees with vertices and -matching number with . When pendent vertices are attached with each certain non-central vertices of , then the resulting graph is denoted by . If we choose , then it means every tree from and contains perfect matching.
Lemma 3. (see [26]). If an n-vertex tree has perfect matching, then there must exist at least two vertices of degree one with neighbouring vertices of degree two, where .
Lemma 4. (see [26]). If an n-vertex tree has an -matching with , then there must exist a pendent vertex which is not saturated by -matching.
In the following, we will find two theorems which will give extreme values of for all trees in .
Theorem 1. Let , , and be integers and ; then, , where equality meets when .
Proof. Suppose . If the tree is isomorphic to , then . On the other hand, if is not isomorphic to , then we assume that the vertex , i.e., where . Lemma 3 assures that there exist vertices and adjacent by an edge with and . Let . We define . It is clear that . By the definition of , we havewhere , , and for .
Note that ; then, obviously . Then, by the construction of and keeping Lemma 3 in our mind, we can choose and in where and . It is clear that . Let . We set . Similarly, . We define ; then,where , and for .
This implies that . We repeat the above process on the graph again and again and we obtain a sequence of graphs with the relation
For some positive integerp, we have and .
Hence, .
Theorem 2. Suppose that , , and be integers. If , then , where equality meets when .
Proof. We claim that ; then, . If we apply the above-defined process (in previous Theorem 1) on , then we will obtain the expression for some positive integer . Hence, equality meets when .
4. Extremal Values of Variable Sum Exdeg Index for Conjugated Unicyclic Graphs
In this portion of the paper, we will find extreme values for among all the conjugated unicyclic graphs in for . In this concern, we will prove some lemmas which will support our main theorems.
Lemma 5. (see [26]). For any tree from , we find at least one vertex of degree 1 which will be adjacent with a vertex of degree 2, i.e., .
Lemma 6. Suppose that and ; then, , where sign of equality meets when .
Proof. Let ; then, by Lemma 4, we find a pendent vertex in which is not saturated by an -matching of . Obviously, the vertices in are saturated by the maximal matching. This implies that . Assume that ; then, . According to Theorem 2, we haveThe above inequality holds if .
If , thenIf , then we havewhere , and for . Finally, we have .
Lemma 7. Let ; then, equality meets when where .
Proof. Let ; then, by Lemma 4, we find a pendent vertex in which is not saturated by a maximal -matching of . Suppose that . Suppose is a vertex in , i.e., . Define ; then, clearly . With the help of Theorem 1, we have , so we haveIf we show that , then it will be enough for the existence of the expression . Since we know that , is not isomorphic to and . If we assume , then . If we assume , thenwhere and .
Hence, . So, we conclude that , and sign of equality meets when .
We define a set . Remember that represents the connected component having the vertex of the graph .
Lemma 8. (see [26]). Let ; then, for every , or .
Lemma 9. Let such that is minimum if where and is one of the pendent vertices of .
Proof. Suppose with minimum variable sum exdeg index. We also assume that and the vertices and are the neighbouring vertices of the vertex along . Here we consider the expressionWe assume that is the connected component of which does not contain the vertex . We can write the expression, . According to Lemma 8, or . In either situation, there exists the following relation: according to Theorem 2 and Lemma 6. Furthermore, the sign of equality meets iff . Next we will prove that the vertex is one of the pendent vertices of such that . We suppose that , so there must exist two vertices and , i.e., . Then, there must be one edge of or which is not included in -matching. Without loss of generality, suppose that does not belong to the -matching. Let where represents the path with as a pendent vertex of . Define ; it is clear that . By the definition of , we havewhere and .
which contradicts our choice of .
Lemma 10. If , with maximum ; then, for every vertex , there exist which will be isomorphic to or . If is isomorphic to , then will be equal to . If is isomorphic to , then the vertex will be the one end vertex of and will be adjacent to some maximum degree vertex of .
Proof. Let with maximum variable sum exdeg index. We also assume that and the vertices and are the neighbouring vertices of the vertex along . Here we consider the expressionWe assume that is the connected component of which does not contain the vertex . We can write the expression .
According to Theorem 1, Lemma 7, and Lemma 8, we haveorfor being even or odd, respectively. Above two inequalities hold iff and , respectively. Next we will prove that(1)If , then .(2)If , then has the vertex as a pendent vertex that is adjacent to the vertex of maximum degree in .For the proof of (i), we assume that . Let , i.e., . We define . By the definition of ,If , thenwhere , , and .
which contradicts our choice of .
If , thenwhere , and .
which contradicts our choice of .
For the proof of (ii), we will just show that and where .
Since and is isomorphic to , .
Note that any vertex other than the vertex of maximum degree in has the degree 2 or 1. If , this implies that in , there will be a vertex which is not saturated by the maximal matching of . Here a contradiction arises for. This implies . If we assume , then once again we find a vertex in which is not saturated by the maximal matching in and again we will find a contradiction. From the above discussion, the proof is accomplished.
Theorem 3. Suppose ; then, for and the sign of equality meets when where is a pendent vertex of .
Proof. Suppose having minimum . According to Lemma 9, for the minimum , will be isomorphic to for every where . For , the above result holds. Now we discuss the above result for . We have . So, we denote , where . We define . It is clear that ; then, by the definition of ,where , and .
which contradicts our choice of . Hence, the proof of above theorem is finished.
5. Main Result
Theorem 4. Let and ; then, the following results must hold:(1)If , then and the sign of equality meets when is not isomorphic to .(2)If and is odd, then sign of equality meets iff .(3)If and is even, then sign of equality meets iff , where , , and .
Proof. Let , with maximum . According to Lemma 8, we are sure that is isomorphic to or where .
For , we have no graph for which . Whenever , then must be isomorphic to where . According to Theorem 3, we can find a graph such that for which contradicts the choice of . With the help of Lemma 10, the proof of (2) or (3) is satisfied. For |B|, we may have the below cases. Case 1. Let ; then, for any two graphs and , both graphs are not isomorphic to and . According to Theorem 3, we have where , and hence (1) satisfies. Case 2. For , we make the following subcases. Let and . Subcase 2.1. Let for every ; then, we have the following set of vertices: and . Let . Choose . If , we define . If , we define In both above , we have and we have the expression which contradicts our choice of G. Subcase 2.2. Let , i.e., ; here we have ; there must exist a vertex in . We define ; then, clearly . By the definition of , we have Since and , this implies that which is a contradiction to the choice of . Subcase 2.3. Let and . In this concern, two subcases arise. Subcase 2.3.1. Suppose that for some ; since we have , there exists a vertex in . According to Lemma 8, Lemma 3, and Lemma 5, there must exist some adjacent vertices say and in , i.e., and . Let . We define ; then, clearly and where , and . which contradicts our choice of . Subcase 2.3.2. Suppose that for some . Let be a vertex in with , so this implies that where is a positive integer. Since we have , then there exists some vertex in . According to Lemma 8, Lemma 3, and Lemma 5, there must exist some adjacent vertices say and in , i.e., and . Let .Remaining portion of the under discussion subcase is similar to subcase 2.3.1 and once again we find the contradiction. According to all above discussion and argument, we follow the desired result.
Theorem 5. Let , , and for every ; then,(1)If and is odd, then sign of equality meets iff .(2)If and is even, then sign of equality meets iff , where , , and .
6. Extremal Values of Variable Sum Exdeg Index for Bicyclic Graphs
Here we are going to define some notations. Let be the collection of bicyclic graphs or all those graphs which have vertices and number of edges. Note that if , then there exist two cycles say and in .(i) is the collection of graphs in which cycles and share a single vertex only.(ii) is the collection of graphs in which cycles and share no common vertex.(iii) is the collection of graphs in which cycles and share a common path of length .
6.1. Extremal Graphs in
Suppose is a graph from the collection , i.e., there are pendent vertices adjacent to a common vertex of and as shown in Figure 3.
Lemma 11. Let ; if , then for .
Proof. Let ; then, by Lemma 2, we obtain another graph say for which . Further by Lemma 1, the graph can be changed into another graph say in which pendent vertices will be attached with some common vertex , of and . If is a not a common vertex of and , then . By the definition of , we have : for , we have where , . : for , we havewhere , .
From the above two cases, we conclude that .
Lemma 12. Let ; then, , , . , , .
Proof. By the definition of , we havewhere and .
Proof of (ii) is the same as proof of (i).
Theorem 6. If , then will be maximal if and for all , the graph from with maximum is .
Proof. Proof of this theorem can be obtained by Lemma 11 and Lemma 12.
6.2. Extremal Graphs in
Here we define that is a graph which is obtained by joining and by a path of length and the remaining number of vertices are attached to the same end vertex of as shown in Figure 4.
Lemma 13. Let ; if , then for .
Proof. Let ; then, by Lemma 2, we obtain another graph say for which . Further by Lemma 1, the graph can be changed into another graph say in which pendent edges are attached with same vertex , i.e., . If is not end vertex of path , then we will show . By the definition of , we havewhere , , and .
Lemma 14. Let ; then, , , . , , . , , .
Proof. By the definition of ,where , , and . This implies that .
Proof of (ii) is the same as proof of (i).
Proof. By the definition of ,where , . This implies that .
. After proving Lemma 13 and Lemma 14, we are able to present the following theorem.
Theorem 7. If , then will be maximal if and for all , the graph from with maximum is .
6.3. Extremal Graphs in
Here we define that is a graph which is obtained by attaching edges to one of the vertices of degree 3 in (see Figure 5). Here we define some lemmas but skip their proofs. We refer Lemma 13 and Lemma 14 for the proof of following lemmas.
Lemma 15. Let ; if , then for .
Lemma 16. Let ; then, , . , . , .
Theorem 8. For and the graph from the collection with maximum for all , and is .
Theorem 9. A graph has maximum variable sum exdeg index if and only if for .
Proof. Since , , and belong to . All the previous lemmas and theorems make it very clear and easy to understand that , , and have maximum , and , , and belong to , , and , respectively, for . Now we just need to compare the of , , and .This implies that .where . This implies that . From the above discussion, we conclude that .
7. Conclusion
Ascertaining the upper and lower bounds on any molecular structure descriptor with regard to various graph parameters is a significant job. We have sought the maximum value of for unicyclic graphs. Sharp bounds have also been investigated for conjugated trees and conjugated unicyclic graphs. We also investigated the extremal graphs for each upper and lower bounds. Following are the main points of conclusion.(i)We have provided maximum and minimum values of for conjugated trees.(ii)We have also provided lower and upper bounds of for unicyclic conjugated graphs with respect to the length of this cycle.(iii)At the end of this paper, we have determined the maximum value of for bicyclic graphs or .
Data Availability
The data used to support the findings of this study are included within the article. However, the reader may contact the corresponding author for more details of the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.