Abstract
The bond incident degree (BID) indices can be written as a linear combination of the number of edges with end vertices of degree and . We introduce two transformations, namely, linearizing and unbranching, on catacondensed pentagonal systems and show that BID indices are monotone with respect to these transformations. We derive a general expression for calculating the BID indices of any catacondensed pentagonal system with a given number of pentagons, angular pentagons, and branched pentagons. Finally, we characterize the CPSs for which BID indices assume extremal values and compute their BID indices.
1. Introduction
A pentagonal system is a connected geometric figure obtained by concatenating congruent regular pentagons side to side in a plane in such a way that the figure divides the plane into one infinite (external) region and a number of finite (internal) regions, and all internal regions must be congruent regular pentagons. In a pentagonal system, two pentagons are adjacent if they share an edge. In this work, we consider only catacondensed pentagonal systems (CPSs), i.e., pentagonal systems which have no internal vertices. This is equivalent to say that pentagonal system has only pentagons of type , and (see Figure 1). We will denote by , and , respectively, the number of , and pentagons the CPS has. We will denote the set of all CPSs with pentagons by . For more details on pentagonal systems and this theory, we refer the readers to [1]. We are interested in studying the behavior of general bond incident degree (BID) indices over pentagonal systems. For details about the pentagonal systems, see [2–7]. One of the much studied topological indices was put forward by the Randić index [8]. For a graph , it is denoted by and is defined aswhere denotes the degree of the vertex and sum runs over all edges of . There are many topological indices which appeared in the literature of mathematical chemistry (see for example [9–13]). Among the degree-based topological descriptors, the most studied are the first and second Zagreb indices [14–17], the sum-connectivity index [17–19], the atom-bond connectivity index [17, 20], the augmented Zagreb index [17, 21, 22], the geometric arithmetic index [23–25], and the harmonic index [19, 26, 27]. Each of these BID indices can be expressed [28] aswhere is the number of edges in with end vertices of degree and and is a finite sequence of nonnegative real numbers. For instance, if BID is the Randić index, then . From now on, BID indices of any graph induced by the sequence are defined by equation (2). In case of CPS, we have only vertices of degree 2, 3, and 4; therefore the general BID indices over will be induced by a sequence of nonnegative real numbers:for every . Now, following [29], we define two transformations on a CPS, called linearizing and unbranching transformations. We will show that BID indices are monotone with respect to these transformations. This will give us a reduction procedure to compute the BID indices of any CPS. Finally, the results obtained will be applied to find the minimal and maximal values of BID indices over .
2. Linearizing and Unbranching a Catacondensed Pentagonal System
In this section, we define two transformations: the linearizing transformation and the unbranching transformation , for the catacondensed pentagonal systems. In order to define these two transformations, we first recall the definition of coalescence of two graphs [30]. Suppose and are graphs with edges and . The coalescence of and with respect to edges and denoted by is formed by identifying the edges and in the following way:and the two edges in are adjacent if(i)These are adjacent in or , or(ii)One is and the other one is adjacent to in or to in , or(iii)One is and the other one is adjacent to in or in , or(iv)First one is and the second one is
Now, we define the linearizing transformation on a catacondensed pentagonal system. Let be catacondensed pentagonal system with angular pentagon and sub-catacondensed pentagonal systems and as shown in Figure 2. Let be the catacondensed pentagonal system obtained by applying coalescing operation on that is moving the catacondensed pentagonal systems to the edge of angular pentagon . We call the pentagonal system P2 the linearization of pentagonal system P1 at the pentagon .
Let ; then, the difference of BID indices of the catacondensed pentagonal systems and is calculated in the next theorem.
Theorem 1. Let be a CPS obtained from by linearizing at angular pentagon . Then,
Proof. In Figure 2, bold edges in and are the ones whose degree will change after the linearizing transformation. We collect all the bold edges in the set from . Similarly, collect all the bold edges in the set from . Note that the set of edges and is in one to one correspondence in such a way that for each edge in , the degree of end vertices is equal to those of corresponding edge in . Hence,If , then from above theorem, we have . Hence, by applying a linearizing transformation, we can construct a new catacondensed pentagonal system whose BID indices are greater than the BID indices of original catacondensed pentagonal system. This fact is reflected in the next example.
Example 1. The sequence of CPS in Figure 3 satisfieswhen .
Next, we define the unbranching transformation on catacondensed pentagonal system. Let be CPS with branched pentagon . Suppose that is a pentagonal chain in . Note that such branched pentagon always exists in . Let be a catacondensed pentagonal system obtained by applying coalescing operation on that is moving the sub-catacondensed pentagonal system to any of the 2-2 edge in the last pentagon of . This process is shown in Figure 4. Let ; then, the difference of BID indices of the catacondensed pentagonal system and can be expressed as the difference of and . This is proved in the next theorem.
Theorem 2. Let be a catacondensed pentagonal system obtained from by unbranching at ; then,
Proof. In Figure 4, bold edges in and are the ones whose degree will change after the unbranching transformation. We collect all the bold edges in the set from . Similarly, collect all the bold edges in the set from . Note that the set of edges and is in one to one correspondence in such a way that for each edge in , the degree of end vertices is equal to those of the corresponding edge in . Hence,Note that if , then each of the BID indices defined on the catacondensed pentagonal system is monotone decreasing when we apply unbranching transformation. Next example depicts this fact.
Example 2. The sequence of CPS in Figure 5 satisfieswhen .
3. Method to Compute BID Indices of Catacondensed Pentagonal System
Examples 1 and 2 show that we can transform any catacondensed pentagonal system into linear pentagonal chain by successively applying linearizing and unbranching transformations. The number of steps depends on the number of angular pentagons and number of branched pentagons . Theorems 1 and 2 show that we can find the exact value of topological indices after applying these transformations. Denote by a linear pentagonal chain with pentagons; then, the topological indices of can be computed by the following formula:
Our next results show that once we know the number of angular pentagons , the number of branched pentagons , and the number of pentagons in a catacondensed pentagonal system, we can compute its topological indices.
Lemma 1. If is a pentagonal chain with angular pentagons, then
Proof. We will prove it by induction on . If , then and the result follows. Suppose that the result holds valid for pentagonal chain with less than pentagons. Let be a pentagonal chain with angular pentagons and let be the pentagonal chain obtained by applying linearizing transformation on an angular pentagon of . Then, by Theorem 1,Since the pentagonal chain has angular pentagons, then by induction hypothesis,Finally,Our next theorem generalizes Lemma 1 to any catacondensed pentagonal system.
Theorem 3. Let be a catacondensed pentagonal system with pentagons, angular pentagons, and branched pentagons; then,
Proof. We will prove it by induction on . If , then by Lemma 1, the result holds. Suppose that the result is true for catacondensed pentagonal systems with less than branched pentagons and let be a catacondensed pentagonal system with branched pentagons. By applying unbranching transformation at branch of , we obtain a pentagonal system which has branched pentagons and angular pentagons. Hence, by Theorem 2,By induction,Hence,Next example details the computation of BID topological indices of any catacondensed pentagonal system using Theorem 3.
Example 3. We compute the topological indices of catacondensed pentagonal system shown in Figure 1. It contains 1 angular pentagon, 1 branched pentagon, and in total 10 pentagons. Hence, by Theorem 3, we haveFor example, if the BID topological index is the Randić index, then and soIf the BID index is the first Zagreb index , then , and
Corollary 1. (i)If and , then the minimal value of BID indices over is attained in .(ii)If and , then the maximal value of BID indices over is attained in .
Proof. Suppose that and ; then, by Theorem 3, the minimum value of BID indices is attained when and . Hence, our catacondensed pentagonal system is a linear pentagonal system. Similarly, when and , then the maximum value is attained when and which occurs in the linear pentagonal system.
From Theorem 3, it is clear that the maximum and the minimum values of BID indices over depend on the values of , , , and . Also, for any catacondensed pentagonal system with , we have and . Let and denote the CPSs with pentagons as shown in Figures 6 and 7.
Corollary 2. (i)If and , then the minimal value of BID indices over is attained in .(ii)If and , then the maximal value of BID indices over is attained in .
Proof. (i) Suppose that and ; then, by Theorem 3, the minimum value of BID indices is attained when . Also, the maximum value of can be which occurs in .
(ii) By Theorem 3, we get the maximum value of BID indices when .
In a catacondensed pentagonal system, we haveWe deduceIf is even (odd), then the maximum value of will be obtained when () and this occurs in .
Corollary 3. (i)If and , then the minimal value of BID indices over is attained in .(ii)If and , then the maximal value of BID indices over is attained in .
Example 4. (i)For the second Zagreb index, we have = . Then, and . Hence, from Corollary 1, the minimum value of BID indices is attained in .(ii)If BID indices are among the Randić, geometric arithmetic, harmonic, and sum-connectivity indices, then and . Hence, from Corollary 2, the maximum value of BID indices is attained in and the minimum value is attained in .(iii)For the atom-bond connectivity index, we have . In this case, and ; hence, from Corollary 1, the maximum value of BID indices is attained in .In Table 1, we have computed the values of and for different BID indices. This helps us to find the CPS which has the maximum and minimum values of the following BID indices.
4. Concluding Remarks
Among the well-known topological indices, there are various bond incident degree (BID) indices. In this paper, we have studied these BID indices for the catacondensed pentagonal systems and derived a general expression for calculating the BID indices of any catacondensed pentagonal system with a given number of pentagons, angular pentagons, and branched pentagons. We have also characterized the systems having maximum/minimum well-known BID indices from the class of all catacondensed pentagonal systems with a fixed number of pentagons. The present study can be extended in several directions. One of such directions is to study the general pentagonal systems for the BID indices. Also, there are many BID indices (see, for example, Table 1) for which we have not been able to characterize the systems having maximum/minimum values from the class of all catacondensed pentagonal systems with a fixed number of pentagons.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Key R&D Program of China (grant no. 2019YFA0706402) and the Natural Science Foundation of Guangdong Province (grant no. 2018A0303130115).