# Bond Incident Degree Indices of Catacondensed Pentagonal Systems

Authors: Yongsheng Rao; Adnan Aslam; Muhammad Unfowan Noor; A. Othman Almatroud; Zehui Shao
Journal: Complexity
 (2020)

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## Abstract

The bond incident degree (BID) indices can be written as a linear combination of the number of edgesxi,j with end vertices of degree i and j. 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.


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## Body

## 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 typeL1,L2,A2, and A3 (see Figure 1). We will denote by l1,l2,a2, and a3, respectively, the number of L1,L2,A2, and A3 pentagons the CPS has. We will denote the set of all CPSs with n pentagons by CHn. 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 G, it is denoted by RG and is defined as(1)RG=∑uvdudv−1/2,where du denotes the degree of the vertex u and sum runs over all edges uv of G. 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] as(2)BIDG=∑1≤i≤j≤Δθi,jxi,j,Δis the maximum degree ofG,where xi,j is the number of edges in G with end vertices of degree i and j and θi,j is a finite sequence of nonnegative real numbers. For instance, if BID is the Randić index, then θi,j=1/ij. From now on, BID indices of any graph G induced by the sequence θi,j 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 CHn will be induced by a sequence θ22,θ23,θ24,θ33,θ34 of nonnegative real numbers:(3)BIDG=θ22x22+θ23x23+θ24x24+θ33x33+θ34x34,for every G∈CHn. 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 CHn.Figure 1
Types of pentagons.


## 2. Linearizing and Unbranching a Catacondensed Pentagonal System

In this section, we define two transformations: the linearizing transformationT1 and the unbranching transformation T2, for the catacondensed pentagonal systems. In order to define these two transformations, we first recall the definition of coalescence of two graphs [30]. Suppose G1 and G2 are graphs with edges u1v1∈EG1 and u2v2∈EG2. The coalescence of G1 and G2 with respect to edges u1v1 and u2v2 denoted by G is formed by identifying the edges u1v1 and u2v2 in the following way:(4)VG=VG1−u1,v1∪VG2−u2,v2∪u,v,and the two edges in G are adjacent if(i)
These are adjacent inG1 or G2, or(ii)
One isu and the other one is adjacent to u1 in G1 or to u2 in G2, or(iii)
One isv and the other one is adjacent to v1 in G1 or v2 in G2, or(iv)
First one isu and the second one is vNow, we define the linearizing transformation on a catacondensed pentagonal system. LetP1 be catacondensed pentagonal system with angular pentagon P and sub-catacondensed pentagonal systems Q1 and Q2 as shown in Figure 2. Let P2 be the catacondensed pentagonal system obtained by applying coalescing operation on P1 that is moving the catacondensed pentagonal systems Q2 to the 2−2 edge of angular pentagon P. We call the pentagonal system P2 the linearization of pentagonal system P1 at the pentagon P.Figure 2
The linearization transformationT1.LetΘ1=−θ22+2θ23−2θ24+3θ33−2θ34; then, the difference of BID indices of the catacondensed pentagonal systems P1 and P2 is calculated in the next theorem.Theorem 1.
LetP2 be a CPS obtained from P1 by linearizing at angular pentagon P. Then,(5)BIDP2−BIDP1=Θ1.Proof.
In Figure2, bold edges in P1 and P2 are the ones whose degree will change after the linearizing transformation. We collect all the bold edges in the set E from P1. Similarly, collect all the bold edges in the set F from P2. Note that the set of edges EP1∖E and EP2∖F is in one to one correspondence in such a way that for each edge in EP1∖E, the degree of end vertices is equal to those of corresponding edge in EP2∖F. Hence,(6)BIDP2−BIDP1=4θ23+3θ33−θ22+2θ23+2θ24+2θ34=−θ22+2θ23−2θ24+3θ33−2θ34=Θ1.
IfΘ1<0, then from above theorem, we have BIDP2<BIDP1. 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 Figure3 satisfies(7)BIDP1>BIDP2>BIDP3>BIDP4>BIDP5>BIDP6,when Θ1<0.Figure 3
BIDP1>BIDP2>BIDP3>BIDP4>BIDP5>BIDP6.Next, we define the unbranching transformation on catacondensed pentagonal system. LetZ1 be CPS with branched pentagon R. Suppose that Q2 is a pentagonal chain in Z1. Note that such branched pentagon always exists in Z1. Let Z2 be a catacondensed pentagonal system obtained by applying coalescing operation on Z1 that is moving the sub-catacondensed pentagonal system Q3 to any of the 2-2 edge in the last pentagon of Q2. This process is shown in Figure 4. Let Θ2=θ22−2θ23+θ33; then, the difference of BID indices of the catacondensed pentagonal system Z1 and Z2 can be expressed as the difference of Θ1 and Θ2. This is proved in the next theorem.Figure 4
The unbranching transformationT2.Theorem 2.
LetZ2 be a catacondensed pentagonal system obtained from Z1 by unbranching at R; then,(8)BIDZ2−BIDZ1=Θ1−Θ2.Proof.
In Figure4, bold edges in Z1 and Z2 are the ones whose degree will change after the unbranching transformation. We collect all the bold edges in the set E from Z1. Similarly, collect all the bold edges in the set F from Z2. Note that the set of edges EZ1∖E and EZ2∖F is in one to one correspondence in such a way that for each edge in EZ1∖E, the degree of end vertices is equal to those of the corresponding edge in EZ2∖F. Hence,(9)BIDZ2−BIDZ1=5θ23+3θ33−2θ22+θ23+2θ34+2θ24+θ33=−2θ22+4θ23−2θ24+2θ33−2θ34=−θ22+2θ23−2θ24+3θ33−2θ34−θ22−2θ23+θ33=Θ1−Θ2.
Note that ifΘ1−Θ2<0, 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 Figure5 satisfies(10)BIDP1>BIDP2>BIDP3,when Θ1−Θ2<0.Figure 5
BIDP1>BIDP2>BIDP3.


## 3. Method to Compute BID Indices of Catacondensed Pentagonal System

Examples1 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 a2 and number of branched pentagons a3. Theorems 1 and 2 show that we can find the exact value of topological indices after applying these transformations. Denote Ln by a linear pentagonal chain with n pentagons; then, the topological indices of Ln can be computed by the following formula:(11)BIDLn=4θ22+2nθ23+2n−3θ33.Our next results show that once we know the number of angular pentagonsa2, the number of branched pentagons a3, and the number of pentagons n in a catacondensed pentagonal system, we can compute its topological indices.Lemma 1.
IfK is a pentagonal chain with a2 angular pentagons, then(12)BIDK=BIDLn−a2Θ1.Proof.
We will prove it by induction ona2. If a2=0, then K=Ln and the result follows. Suppose that the result holds valid for pentagonal chain with less than a2 pentagons. Let K be a pentagonal chain with a2>0 angular pentagons and let L be the pentagonal chain obtained by applying linearizing transformation on an angular pentagon K1 of K. Then, by Theorem 1,(13)BIDL−BIDK=Θ1.
Since the pentagonal chainL has a2−1 angular pentagons, then by induction hypothesis,(14)BIDL=BIDLn−a2−1Θ1.
Finally,(15)BIDK=BIDL−Θ1=BIDLn−a2−1Θ1−Θ1=BIDLn−a2Θ1.
Our next theorem generalizes Lemma1 to any catacondensed pentagonal system.Theorem 3.
LetM be a catacondensed pentagonal system with n pentagons, a2 angular pentagons, and a3 branched pentagons; then,(16)BIDM=BIDLn−a2Θ1−a3Θ1−Θ2.Proof.
We will prove it by induction ona3. If a3=0, then by Lemma 1, the result holds. Suppose that the result is true for catacondensed pentagonal systems with less than a3>0 branched pentagons and let M be a catacondensed pentagonal system with a3 branched pentagons. By applying unbranching transformation at branch M1 of M, we obtain a pentagonal system N which has a3−1 branched pentagons and a2 angular pentagons. Hence, by Theorem 2,(17)BIDN−BIDM=Θ1−Θ2.
By induction,(18)BIDN=BIDLn−a2Θ1−a3−1Θ1−Θ2.
Hence,(19)BIDM=BIDN−Θ1+Θ2=BIDLn−a2Θ1−a3−1Θ1−Θ2−Θ1+Θ2=BIDLn−a2Θ1−a3Θ1−Θ2.
Next example details the computation of BID topological indices of any catacondensed pentagonal system using Theorem3.Example 3.
We compute the topological indices of catacondensed pentagonal systemP1 shown in Figure 1. It contains 1 angular pentagon, 1 branched pentagon, and in total 10 pentagons. Hence, by Theorem 3, we have(20)BIDP1=BIDL10−Θ1−Θ1−Θ2=BIDL10−2Θ1+Θ2=4θ22+16θ23+13θ33−2−θ22+2θ23−2θ24+3θ33−2θ34+θ22−2θ23+θ33=7θ22+10θ23+4θ24+8θ33+4θ34.
For example, if the BID topological index is the Randić index, thenθij=1/ij and so(21)RP1=712+1016+418+813+4112=14.7701.
If the BID index is the first Zagreb indexM1P1, then θij=i+j, and(22)M1P1=7×4+10×5+4×8+8×6+4×7=186.Corollary 1.
(i)
IfΘ1<0 and Θ1−Θ2<0, then the minimal value of BID indices over CPn is attained in Ln.(ii)
IfΘ1>0 and Θ1−Θ2>0, then the maximal value of BID indices over CPn is attained in Ln.Proof.
Suppose thatΘ1<0 and Θ1−Θ2<0; then, by Theorem 3, the minimum value of BID indices is attained when a2=0 and a3=0. Hence, our catacondensed pentagonal system is a linear pentagonal system. Similarly, when Θ1>0 and Θ1−Θ1>0, then the maximum value is attained when a2=0 and a3=0 which occurs in the linear pentagonal system.
From Theorem3, it is clear that the maximum and the minimum values of BID indices over CPn depend on the values of a2, a3, Θ1, and Θ2. Also, for any catacondensed pentagonal system CPn with n≥2, we have l1≥2 and 0≤l2,a2,a3≤n−2. Let En and Fn denote the CPSs with n pentagons as shown in Figures 6 and 7.Figure 6
Catacondensed pentagonal systemEn for n=7.Figure 7
Catacondensed pentagonal systemFn for n = odd and n = even.Corollary 2.
(i)
IfΘ1>0 and Θ1−Θ2<0, then the minimal value of BID indices over CPn is attained in En.(ii)
IfΘ1>0 and Θ1−Θ2<0, then the maximal value of BID indices over CPn is attained in Fn.Proof.
(i) Suppose thatΘ1>0 and Θ1−Θ2<0; then, by Theorem 3, the minimum value of BID indices is attained when a3=0. Also, the maximum value of a2 can be n−2 which occurs in En.
(ii) By Theorem3, we get the maximum value of BID indices when a2=0.
In a catacondensed pentagonal system, we have(23)l1=a3+2,n=a2+l2+a3+l1.
We deduce(24)a3=12n−a2+l2+2=12n−l2+2.
Ifn is even (odd), then the maximum value of a3 will be obtained when l2=0 (l2=1) and this occurs in Fn.Corollary 3.
(i)
IfΘ1<0 and Θ1−Θ2>0, then the minimal value of BID indices over CPn is attained in Fn.(ii)
IfΘ1<0 and Θ1−Θ2>0, then the maximal value of BID indices over CPn is attained in En.Example 4.
(i)
For the second Zagreb index, we haveθij = ij. Then, Θ1=−5<0 and Θ1−Θ2=−6<0. Hence, from Corollary 1, the minimum value of BID indices is attained in Ln.(ii)
If BID indices are among the Randić, geometric arithmetic, harmonic, and sum-connectivity indices, thenΘ1>0 and Θ1−Θ2<0. Hence, from Corollary 2, the maximum value of BID indices is attained in Fn and the minimum value is attained in En.(iii)
For the atom-bond connectivity index, we haveθij=i+j−2/i+j. In this case, Θ1>0 and Θ1−Θ2>0; hence, from Corollary 1, the maximum value of BID indices is attained in Ln.
In Table1, we have computed the values of Θ1 and Θ2 for different BID indices. This helps us to find the CPS which has the maximum and minimum values of the following BID indices.Table 1
Values ofΘ1 and Θ2 for different BID indices.
BID indicesi+jij1/ij2ij/i+j2/i+j1/i+jij3/i+j−23i+j−2/i+jΘ1−2−50.0320.09440.06190.0467−1.4760.0019Θ2010.1680.4040.3330.1383.3906−0.404MaximalEnEnEnEnLnMinimalLnLnFnFnFnFnLn


## 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, Table1) 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.



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