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A numerical quantity TI associated with a graph G satisfying the equation TI(G)=TI(G′) for every graph G′ isomorphic to G is called a graph invariant. In chemical graph theory, graph invariants that are applied in chemical investigations are known as topological indices. The goal of defining a topological index is to associate each chemical structure with a numerical value and thus investigate its properties. In fact, topological indices have found applications in Chemistry [1,2], Computational Linguistics [3], Ecology [4]. Nowadays, a vast number of topological indices exist in the literature [5]. In this paper, we pay our attention to only degree-based topological indices; whose general form is:
TI(G)=∑vu∈E(G)f(dv,du), | (1.1) |
where f is some real valued function with the property f(x,y)=f(y,x) for x,y∈{1,2…} and dv is the degree of a node v∈V(G). In the development of applications, degree-based topological indices have become a powerful tool, for instance, Forgotten index (f(x,y)=x2+y2) reflects the structure-dependency of total π-electron energy Eπ and measures the physical-chemical properties of molecular structures [6,7], the GA index (f(x,y)=2√xyx+y) can be used as predictive tool in QSPR/QSAR researches [8] and the atom-bond connectivity index (f(x,y)=√x+y−2xy) has proven to be a valuable predictive index in study of heat of formation in alkanes [9].
On the other hand, a polyomino system is a finite 2-connected plane graph such that each interior face (say a cell) is surrounded by a regular square of length one. In a polyomino system, two squares are said to be adjacent if they share a side. A polyomino chain is a polyomino system in which the joining of the centres of its adjacent cells forms a path c1c2…cn, where ci is the centre of the ith cell. Hence, in a polyomino chain every square is adjacent with at most two other squares. If a square has only one adjacent square, it is called terminal, if it has two adjacent squares having no vertex of degree 2, it is called medial, and if it has two adjacent squares such that it has a vertex of degree 2, it is called kink. A polyomino chain without kinks is called linear chain Lin. A polyomino chain consisting of only kinks and terminal squares is known as zigzag chain Zn (see Figure 1). A maximal linear chain (containing the terminal squares and kinks at its end) in the polyomino chains is called a segment of the polyomino chain.
The name polyomino was introduced in 1953 in analogy to dominoes by Solomon W. Golomb [10] and since then polyomino systems have been widely studied, as a matter of fact, in organic chemistry, especially in polycyclic aromatic compounds. At the present time, recent works on the polyomino chains include perfect matchings [11,12], finding formulas for calculating several topological indices [13,14,15,16] and extremal problems [17,18,19,20,21,22]. Specifically, random polyomino chains have attracted substantial attention from researchers in recent years [23,24,25,26,27].
A random polyomino chain (RPCn=RPC(n,p1,p2)) could be constructed by the following way: for n=1 and n=2, RPCn are shown in Figure 2. For n≥3, a new square can be attached in two ways, which results in RPC1n and RPC2n with probability p1 and p2 respectively, where 0<p1,p2<1 and p1+p2=1, see Figure 3. For a random polyomino chain at time n, the value of a topological index is a random variable. Considering the arguments put forward in the previous paragraphs and by using a martingale approach, in this paper, we establish an asymptotic distribution for degree-based topological indices in a random polyomino chain. Moreover, their explicit analytical expressions of the expected value and variance are obtained. As a result, we show a general expression for calculating the degree-based topological indices for a polyomino chain. Finally, we compute the expected value of several degree-based topological indices, such as, Sombor, Forgotten, Zagreb index of a random polyomino chain.
In this section, we state and prove our main results. First, let Ln denote the link selected at time n≥3, i.e., Ln denotes a random variable with range {1,2} where pi=P(Ln=i). For i, j∈{1,2}, TIn=TI(RPCn), RPCin denotes a random polyomino chain at time n≥3 such that Ln=i, TIn,i=TI(RPCin), RPCj,in denotes a random polyomino chain at time n≥4 such that Ln−1=j and Ln=i, TIn,j,i=TI(RPCj,in), αj,i=TI4,j,i−TI3,j, αi=TI3,i−TI2, α=∑2j=1∑2i=1αj,ipjpi and β=∑2j=1∑2i=1α2j,ipjpi.
Remark 1. Note that, by definition:
1. α1,1=α1=3f(3,3),
2. α1,2=3f(3,4)+f(2,4)+f(2,3)−2f(3,3),
3. α2,1=f(3,4)−f(2,4)+f(2,3)+2f(3,3),
4. α2,2=f(4,4)+2f(2,4),
5. α2=2f(3,4)+2f(2,4)−f(3,3).
Then, α2,1−α1,2=α1,1−α2. In particular, when f(x,y)=xa+ya with a∈R and x,y∈{1,2,…} the following conditions are satisfied:
1. α1,1=α2,1,
2. α2,2=α1,2=α2.
Besides, in this case, TIn=∑v∈V(RPCn)(dv)a+1, due to the following identity
∑vu∈E(G)(dv)a+(du)a=∑v∈V(G)(dv)a+1, |
the validity of the previous expression can be consulted, for instance in [28].
Theorem 1. Let RPCn=RPC(n,p1,p2) be a random polyomino chain, then for n≥3
E(TIn)=E(TI3)+α(n−3), |
V(TIn)=V(TI3)+(β−α2)(n−3), |
where
E(TI3)=TI2+2∑i=1αipi, |
V(TI3)=2∑i=1α2ipi−(2∑i=1αipi)2. |
Proof. For n≥4, it follows from the definition of a random polyomino chain and by the definition of TI(G) in Equation (1.1) the following almost-sure recursive relation of TIn conditional on Fn−1 and the random vector (Ln−1,Ln)
TIn,Ln−1,Ln−TIn−1=TI4,Ln−1,Ln−TI3,Ln−1, |
where Fn−1 denotes the σ-field generated by the history of the growth of the random polyomino chain in the first n−1 stages. Now for n≥4, we take the expectation with respect to (Ln−1,Ln) to get
E(TIn∣Fn−1)=2∑j=12∑i=1(TIn−1+αj,i)pjpi=TIn−1+2∑j=12∑i=1αj,ipjpi, |
where, αj,i=TI4,j,i−TI3,j. Then, taking expectation, we obtain a recurrence relationship for E(TIn) with n≥4,
E(TIn)=E(TIn−1)+2∑j=12∑i=1αj,ipjpi. | (2.1) |
We solve Equation (2.1) with the initial value E(TI3) and we obtain the result stated in the theorem,
E(TIn)=E(TI3)+α(n−3), |
where α=2∑j=12∑i=1αj,ipjpi. For n≥4, the expression for E(TI2n) follows in a similar manner,
E(TI2n∣Fn−1)=2∑j=12∑i=1(TIn−1+αj,i)2pjpi=2∑j=12∑i=1TI2n−1pjpi+2TIn−1αj,ipjpi+α2j,ipjpi=TI2n−1+2αTIn−1+β, |
where β=2∑j=12∑i=1α2j,ipjpi, thus
E(TI2n)=E(TI2n−1)+2αE(TIn−1)+β=E(TI2n−1)+2αE(TI3)+2α2(n−4)+β, |
then iterating, for n≥3 it is obtained that
E(TI2n)=E(TI23)+(2αE(TI3)+β)(n−3)+α2(n−3)(n−4). |
For n≥3, the variance of TIn is obtained immediately by taking the difference between E(TI2n) and E(TIn)2,
V(TIn)=V(TI3)+β(n−3)+((n−3)(n−4)−(n−3)2)α2=V(TI3)+(β−α2)(n−3). |
Finally, note that
E(TI3)=E(E(TI3∣L3))=2∑i=1(TI2+αi)pi=TI2+2∑i=1αipi, |
where αi=TI3,i−TI2. In the same manner, we have
V(TI3)=2∑i=1α2ipi−(2∑i=1αipi)2, |
proving the theorem.
Observe that the following statements are equivalent
1. β−α2=0.
2. α1,1=α1,2=α2,2=α2,1.
3. For n≥2, TIn=TI2+α(n−2) almost surely.
4. f(3,4)=(2f(4,4)+f(2,4))/3, f(3,3)=(f(4,4)+2f(2,4))/3 and f(2,3)=(−f(4,4)+4f(2,4))/3.
Consequently, when β−α2=0, TInn converges almost surely to α as n→∞. It is worth noting that by using the equivalences stated above we can conclude that, TIn is a deterministic sequence almost surely if and only if α1,1=α1,2=α2,2=α2,1. Hence, by Remark 3 if f(x,y)=xa+ya with a∈R we have that TIn is a deterministic sequence almost surely if and only if 2⋅3a+1=4a+1+2a+1,a∈R. The last equation has two unique solutions a=0,−1, since for a∈(−1,0), xa+1 is a strictly concave function on R+ hence (4+22)a+1>4a+1+2a+12 and for a>0 or a<−1, xa+1 is a strictly convex function on R+ hence (4+22)a+1<4a+1+2a+12. Therefore, TIn is a deterministic sequence almost surely if and only if a∈{0,−1}. This fact makes sense since
∑vu∈E(RPCn)(dv)0+(du)0=∑v∈V(RPCn)(dv)1=2|E(RPCn)|=2+6n, |
and
∑vu∈E(RPCn)(dv)−1+(du)−1=∑v∈V(RPCn)(dv)0=|V(RPCn)|=2+2n. |
Now, we exploit a martingale formulation to investigate the asymptotic behavior of TIn when β−α2>0.
Proposition 2. For n≥3, {Mn=TIn−α(n−3)}n is a martingale with respect to Fn.
Proof. Observe that E(|Mn|)<+∞. For n≥4, by Theorem 1,
E(Mn∣Fn−1)=E(TIn−α(n−3)∣Fn−1)=E(TIn∣Fn−1)−α(n−3)=TIn−1+α−α(n−3)=TIn−1−α(n−4)=Mn−1. |
The proof is completed.
We use the notation D⟶ to denote convergence in distribution and P⟶ to denote convergence in probability. Here, N(μ,σ2) denotes a random variable with normal distribution with mean μ and variance σ2.
Theorem 3. As n→∞,
TIn−(n−3)α√nD⟶N(0,β−α2). |
Proof. For k≥4 and j,i∈{1,2}, we have
|∇Mk|=|∇TIk−α|≤2max(j,i){|αj,i|}, |
where ∇Mk=Mk−Mk−1 and ∇TIk=TIk−TIk−1. That is, given ε>0, there exists an N0(ε)>0 such that, the sets {|∇Mk|>ε√n} are empty for all n>N0(ε). Then, we conclude that
Un:=1nn∑k=4E((∇Mk)2I{|∇Mk|>ε√n}∣Fk−1), |
converges to 0 almost surely, hence, UnP⟶0. Then, the Lindeberg's condition is verified. Next, the conditional variance condition is given by
Vn:=1nn∑k=4E((∇Mk)2∣Fk−1)P⟶β−α2. |
Since,
1nn∑k=4E((∇Mk)2∣Fk−1)=1nn∑k=4E((∇TIk−α)2∣Fk−1)=1nn∑k=42∑j=12∑i=1(αj,i−α)2pjpi=n−3n2∑j=12∑i=1(αj,i−α)2pjpi. |
Therefore, by the Martingale Central Limit Theorem [29], we thus obtain the stated result.
Finally, in order to apply the results obtained in this section, we compute the expected value of several important topological indices for a random polyomino chain (see Table 1).
TI | A | B | C | D | E |
first Zagreb index | 0 | -2 | 20 | -2 | -6 |
second Zagreb index | -1 | -4 | 32 | -4 | -24 |
first hyper-Zagreb index | -2 | -26 | 136 | -26 | -106 |
second hyper-Zagreb index | -21 | -120 | 384 | -92 | -560 |
modified first Zagreb index | 0 | -13/144 | 5/16 | -13/144 | 43/72 |
Albertson index | -2 | -2 | 4 | -6 | -2 |
extended index | 1/6 | -2/3 | 7/2 | -7/12 | 5/12 |
sigma index | 2 | -10 | 8 | -10 | -10 |
Sombor index | 395/3349 | -225/113 | 2599/178 | -2102/1065 | -2108/441 |
Randić index | -34/2413 | 184/3229 | 1138/1189 | 224/4583 | 1338/1279 |
reciprocal Randić index | -255/2588 | -426/763 | 985/102 | -509/870 | -665/257 |
sum-connectivity index | -33/2872 | 449/6784 | 461/394 | 382/6307 | 1092/1283 |
reciprocal sum-connectivity index | 302/14565 | -731/1829 | 5216/675 | -42/107 | -59/1243 |
harmonic index | -11/420 | 23/210 | 11/12 | 2/21 | 457/420 |
atom-bond-connectivity (ABC) index | 183/6023 | -432/7583 | 991/489 | -130/3373 | 691/796 |
augmented Zagreb index | -1636/757 | 2399/1751 | 944/27 | 515/269 | -1814/137 |
forgotten index | 0 | -18 | 72 | -18 | -58 |
geometric-arithmetic index | -307/9318 | 353/2396 | 883/306 | 380/2817 | 637/565 |
arithmetic-geometric index | 413/10692 | -365/2282 | 3499/1121 | -976/6871 | 969/1126 |
inverse sum indeg index | -19/210 | -8/105 | 14/3 | -2/21 | -116/105 |
In this section, the goal is to obtain explicit analytical expressions to calculate TI(PCn) where PCn is a polyomino chain with n squares. Let m≥1 and i∈{1,2,…,m} note that a polyomino chain PCn consists of a sequence of segments s1,s2,…,sm (see Figure 4) with lengths l(si)=li such that ∑mi=1li=n+m−1, where li is calculated by the number of squares in si.
Theorem 4. Let PCn be a polyomino chain having n≥3 squares and m≥1 segment(s) si with i=1,2,…,m. Then
TI(PCn)=3f(3,3)n+(4f(3,4)+2f(2,3)−6f(3,3))m+(f(2,4)−f(2,3)+f(3,3)−f(3,4))(I1+Im)+(f(4,4)+2f(2,4)−4f(3,4)−2f(2,3)+3f(3,3))γ+2f(2,2)+2f(2,3)+f(3,3)−4f(3,4), |
where, Ii={1ifli=20ifli≠2 and γ=m−1∑i=2Ii.
Proof. Note that PCn is a realization of RPCn, then we know the value of Lk for k=3,4,…,n. Therefore, by using the ideas presented in Section 2 we have,
TI(PCn)=TI2+α2I1+α1,1(1−I1)+n∑k=4αLk−1,Lk=TI2+(α2−α1,1)I1+α1,1+2∑j=12∑i=1Xj,iαj,i, |
where Xj,i=|{k∈{4,5,…,n}|Lk−1=j and Lk=i in PCn}| and I1=I{l1=2}. Now, if at time k (3≤k≤n), Lk=2 then the last segment in PCk−1 is finished (so, a new segment is initiated in PCk) and if at time k, Lk=1 then a square is added to the last segment in PCk−1. Hence, X2=|{k∈{3,4,…,n}|Lk=2 in PCn}|=m−1 and X1=|{k∈{3,4,…,n}|Lk=1 in PCn}|=n−2−(m−1)=n−m−1. Moreover, X1,2=|{i∈{1,2,…,m−1}|li≠2 in PCn}| and X2,1=|{i∈{2,3,…,m}|li≠2 in PCn}|. We may write this as: X1,2=m−γ−1−I1 and X2,1=m−γ−1−Im, where,
Ii={1ifli=20ifli≠2andγ=m−1∑i=2Ii. |
Consequently, X1,1=n−2m+γ−1+I1+Im and X2,2=γ, because of the following identities
X1,1+X2,1=X1−1+I1, |
X2,2+X1,2=X2−I1. |
Finally, we arrive at the desired result by replacing the values of Xj,i, αj,i and α2.
Remark 2. 1. By using that m∑i=1li=n+m−1, it is verified that, X1,1=∑li≠2(li−3).
2. On the other hand, by definition if f(x,y)=xa+ya with a∈R then the coefficients of γ and I1+Im in Theorem 4 are zero and the coefficient of m is zero, i.e., the general expression showed in Theorem 4 is independent of m if and only if 2⋅3a+1=4a+1+2a+1 if and only if a∈{0,−1}.
3. Finally, by the way, in [30] the authors established a general expression for calculating the bond incident degree (BID) indices of a polyomino chain; which follows from Theorem 4. BID indices form a subclass of the class all degree-based topological indices.
By definition if PCn=Lin, we deduce that m=1 and l1=n and if PCn=Zn, then m=n−1 and li=2 for i=1,2,…,m. Therefore, the following corollary may be obtained directly by Theorem 4.
Corollary 1. Let Lin and Zn be linear and zigzag chains respectively with n≥3 squares. Then
TI(Lin)=3f(3,3)n+4f(2,3)+2f(2,2)−5f(3,3), |
TI(Zn)=(2f(2,4)+f(4,4))n+4f(2,3)−3f(4,4)+2f(3,4)−4f(2,4)+2f(2,2).
It is worth noting that in 2020, Buragohain et al. [31] introduced a novel generalized topological index for some chemical structures defined as
ISI(α,β)(G)=∑uv∈E(G)(d(u)d(v))α(d(u)+d(v))β. |
In [13] the authors studied the generalized ISI(α,β)-index and (α,β)-Zagreb index of a linear chain. By using Corollary 1 the results showed in [13] can be obtained. In addition, taking f(x,y)=x2+y2 in Equation (1.1), we obtain the Forgotten index. Recently, in [15] the computation of the Forgotten index in a polyomino chain was given as follows:
Corollary 2. Let n≥2 and PCn be a polyomino chain with m≥1 segment(s). Then F(PCn)=54n+18m−40.
Note that, the general expression obtained in Corollary 2 is independent of γ,I1 and Im; which makes sense because of Remark 4. In a similar manner, we can obtain the above result from Theorem 4. Finally, in the following results by using Theorem 4 we will compute TI(PCn) of several kinds of polyomino chains.
Corollary 3. For the polyomino chain with n≥3 squares and 2 segments s1 and s2 satisfy l1=2and l2=n−1, PC1n, we have the following:
TI(PC13)=2f(3,4)+4f(2,3)+2f(2,4)+2f(2,2), |
and for n≥4
TI(PC1n)=3f(3,3)n+3f(3,4)+5f(2,3)−10f(3,3)+f(2,4)+2f(2,2). |
Corollary 4. For the polyomino chain with n≥5 squares and m≥3 segments s1,s2,…,sm satisfy l1=lm=2and l2,l3,…,lm−1≥3, PC2n, we have the following:
TI(PC2n)=3f(3,3)n+(4f(3,4)+2f(2,3)−6f(3,3))m+3f(3,3)−6f(3,4)+2f(2,4)+2f(2,2). |
Corollary 5. For the polyomino chain with n≥6 squares and m≥3 segments s1,s2,…,sm satisfy l1=2 and l2,l3,…,lm≥3 or lm=2 and l1,l2,…,lm−1≥3, PC3n, we have the following:
TI(PC3n)=3f(3,3)n+(4f(3,4)+2f(2,3)−6f(3,3))m+2f(2,2)+f(2,3)+2f(3,3)−5f(3,4)+f(2,4). |
Corollary 6. For the polyomino chain with n≥7 squares and m≥3 segments s1,s2,…,sm satisfy l1,l2,…,lm≥3, PC4n, we have the following:
TI(PC4n)=3f(3,3)n+(4f(3,4)+2f(2,3)−6f(3,3))m+2f(2,2)+2f(2,3)+f(3,3)−4f(3,4). |
Actually, the authors in [32,33,34] calculated several topological indices, such as, redefined Zagreb index, harmonic index and inverse sum index for Ln, Zn and PCin with i=1,2; which are deduced from Corollaries 1, 3 and 4. Besides, in [7,35,36] the authors computed Forgotten, {R}andić and generalized Zagreb index for Ln, Zn and PCin with i=1,2,3,4; hence we can deduce the results above mentioned by using Corollaries 1, 3, 4, 5 and 6. In fact the results showed in [7] can be verified directly by Corollary 2.
On the other hand, here a polyomino chain of dimension n≥1 with k=k1+k2+k3 where k1 is the number of kinks, k2 is the number of medials and k3 is the number of terminals in a unit of polyomino chain will be denoted by PCn,k. In Figure 5, a general representation of a polyomino chain PCn,k is depicted. Let k≥3, by definition of PCn,k, we have: m=2n, γ=n−1, Im=1 and I1=I{k=3}. Hence, in the following corollary, we will compute TI(PCn,k) for k≥3 by using Theorem 4.
Remark 3. Note that, by definition PCn,1=Lin and PCn,2=Z2n.
Corollary 7. Let k≥3, n≥1, then we have
TI(PCn,k)=(3(k−3)f(3,3)+4f(3,4)+2f(2,3)+f(4,4)+2f(2,4))n+(f(2,4)−f(2,3)+f(3,3)−f(3,4))I{k=3}+2f(2,2)+3f(2,3)−f(3,3)−f(3,4)−f(2,4)−f(4,4). |
In fact, in [37] Hayat et al. computed the exact analytical expressions of the ABC, GA, ABC4 and GA5 index for PCn,k with k=3,4,5. These results can be obtained as a consequence of Corollary 7.
In this paper, we proposed a martingale approach to the study of topological indices in random polyomino chains. The expected value and variance have been determined and we formulated a martingale to characterize the asymptotic behavior of the topological indices. Moreover, we considered some particular topological indices, such as, the first Zagreb, Sombor, harmonic, geometric-arithmetic and second Zagreb index for a random polyomino chain. In fact, from the derived results, several known results about polyomino chains were obtained as corollaries. We believe the results obtained in this paper can provide theoretical support for the chemical research. By the way, the extremal random polyomino chains with respect to several well-known degree-based topological indices have been discussed in our next paper. Finally, it would be interesting to extend the work of this paper to k-polygonal chains. We expect to develop it in the future.
All three authors would like to thank two anonymous referees for valuable comments that definitely help improve the quality of the paper.
The authors declare there is no conflict of interest.
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TI | A | B | C | D | E |
first Zagreb index | 0 | -2 | 20 | -2 | -6 |
second Zagreb index | -1 | -4 | 32 | -4 | -24 |
first hyper-Zagreb index | -2 | -26 | 136 | -26 | -106 |
second hyper-Zagreb index | -21 | -120 | 384 | -92 | -560 |
modified first Zagreb index | 0 | -13/144 | 5/16 | -13/144 | 43/72 |
Albertson index | -2 | -2 | 4 | -6 | -2 |
extended index | 1/6 | -2/3 | 7/2 | -7/12 | 5/12 |
sigma index | 2 | -10 | 8 | -10 | -10 |
Sombor index | 395/3349 | -225/113 | 2599/178 | -2102/1065 | -2108/441 |
Randić index | -34/2413 | 184/3229 | 1138/1189 | 224/4583 | 1338/1279 |
reciprocal Randić index | -255/2588 | -426/763 | 985/102 | -509/870 | -665/257 |
sum-connectivity index | -33/2872 | 449/6784 | 461/394 | 382/6307 | 1092/1283 |
reciprocal sum-connectivity index | 302/14565 | -731/1829 | 5216/675 | -42/107 | -59/1243 |
harmonic index | -11/420 | 23/210 | 11/12 | 2/21 | 457/420 |
atom-bond-connectivity (ABC) index | 183/6023 | -432/7583 | 991/489 | -130/3373 | 691/796 |
augmented Zagreb index | -1636/757 | 2399/1751 | 944/27 | 515/269 | -1814/137 |
forgotten index | 0 | -18 | 72 | -18 | -58 |
geometric-arithmetic index | -307/9318 | 353/2396 | 883/306 | 380/2817 | 637/565 |
arithmetic-geometric index | 413/10692 | -365/2282 | 3499/1121 | -976/6871 | 969/1126 |
inverse sum indeg index | -19/210 | -8/105 | 14/3 | -2/21 | -116/105 |
TI | A | B | C | D | E |
first Zagreb index | 0 | -2 | 20 | -2 | -6 |
second Zagreb index | -1 | -4 | 32 | -4 | -24 |
first hyper-Zagreb index | -2 | -26 | 136 | -26 | -106 |
second hyper-Zagreb index | -21 | -120 | 384 | -92 | -560 |
modified first Zagreb index | 0 | -13/144 | 5/16 | -13/144 | 43/72 |
Albertson index | -2 | -2 | 4 | -6 | -2 |
extended index | 1/6 | -2/3 | 7/2 | -7/12 | 5/12 |
sigma index | 2 | -10 | 8 | -10 | -10 |
Sombor index | 395/3349 | -225/113 | 2599/178 | -2102/1065 | -2108/441 |
Randić index | -34/2413 | 184/3229 | 1138/1189 | 224/4583 | 1338/1279 |
reciprocal Randić index | -255/2588 | -426/763 | 985/102 | -509/870 | -665/257 |
sum-connectivity index | -33/2872 | 449/6784 | 461/394 | 382/6307 | 1092/1283 |
reciprocal sum-connectivity index | 302/14565 | -731/1829 | 5216/675 | -42/107 | -59/1243 |
harmonic index | -11/420 | 23/210 | 11/12 | 2/21 | 457/420 |
atom-bond-connectivity (ABC) index | 183/6023 | -432/7583 | 991/489 | -130/3373 | 691/796 |
augmented Zagreb index | -1636/757 | 2399/1751 | 944/27 | 515/269 | -1814/137 |
forgotten index | 0 | -18 | 72 | -18 | -58 |
geometric-arithmetic index | -307/9318 | 353/2396 | 883/306 | 380/2817 | 637/565 |
arithmetic-geometric index | 413/10692 | -365/2282 | 3499/1121 | -976/6871 | 969/1126 |
inverse sum indeg index | -19/210 | -8/105 | 14/3 | -2/21 | -116/105 |