Topological indices correlate certain physicochemical properties like boiling point, stability, and strain energy of chemical compounds. In this report, we compute M-polynomials for PAMAM dendrimers and polyomino chains. Moreover, by applying calculus, we compute nine important topological indices of under-study dendrimers and chains.
Dong-A University1. Introduction
The polyomino chains constitute a finite 2-connected floor plan, where each inner face (or a unit) is surrounded by a square of length one. We can say that it is a union of cells connected by edges in a planar square lattice. For the origin of dominoes, we quote [1]. The polyomino chains have a long history dating back to the beginning of the 20th century, but they were originally promoted by Golomb [2, 3]. Dendrimers [4] are repetitively branched molecules. The name comes from the Greek word, which translates to “trees.” Synonymous terms for dendrimers include arborols and cascade molecules. The first dendrimer was made by Fritz Vögtle in [5]. For detailed study about dendrimer structures we refer the reader to [6–9].
Many studies have shown that there is a strong intrinsic link between the chemical properties of chemical compounds and drugs (such as melting point and boiling point) and their molecular structure [10, 11]. The topological index defined on the structure of these chemical molecules can help researchers better understand the physical characteristics, chemical reactivity, and biological activity [12]. Therefore, the study of topological indices of chemical substances and chemical structures of drugs can make up for the lack of chemical experiments and provide theoretical basis for the preparation of drugs and chemical substances. In the previous two decades, a number of topological indices have been characterized and utilized for correlation analysis in pharmacology, environmental chemistry, toxicology, and theoretical chemistry [13]. Hosoya polynomial (Wiener polynomial) [14] plays a pivotal role in finding topological indices that depend on distances. From this polynomial, a long list of distance-based topological indices can be easily evaluated. A similar breakthrough was obtained recently by Klavžar et al. [15], in the context of degree-based indices. In the year 2015, authors in [15] introduced the M-polynomial, which plays similar role “to what Hosoya polynomial does” to determine many topological indices depending on the degree of end vertices [16–20].
In the present paper, we compute M-polynomials for different dendrimer structures and polyomino chains. By applying fundamental calculus, we recover nine degree-based topological indices for these dendrimers and chains.
2. Basic Definitions and Literature Review
In this paper, we fixed G as a connected graph, V (G) is the set of vertices, E (G) is the set of edges, and dv is the degree of any vertex v. Most of the definitions presented in this section can be found in [17].
Definition 1 (see [15]).
The M-polynomial of G is defined as(1)MG;x,y=∑δ≤i≤j≤ΔmijGxiyjwhere δ=mindv∣v∈VG, Δ=maxdv∣v∈VG, and mij(G) is the edge vu∈E(G) that is i≤j.
The very first topological index was the Wiener index, defined by Wiener in 1945, when he was studying boiling point of alkane [21]. For comprehensive details about the applications of Wiener index, see [22, 23]. After that, in 1975, Milan Randić [24] introduced the first degree-based topological index, which is now known as Randić index and is defined as(2)R-1/2G=∑uv∈EG1dudv.The generalized Randić index is defined as(3)RαG=∑uv∈EG1dudvα;please see [25–29].
The inverse generalized Randić index is defined as(4)RRαG=∑uv∈EGdudvα.It can be seen easily that the Randić index is particular case of the generalized Randić index and the inverse generalized Randić index. Other oldest degree-based topological indices are Zagreb indices. The first Zagreb index is defined as(5)M1G=∑uv∈EGdu+dvand the second Zagreb index is defined as(6)M2G=∑uv∈EGdu×dv.The second modified Zagreb index is defined as(7)Mm2G=∑uv∈EG1dudv.For detailed study about Zagreb indices, we refer the reader to [30–32]. There are many other degree-based topological indices, for example, symmetric division index: (8)SDDG=∑uv∈EGmindu,dvmaxdu,dv+maxdu,dvmindu,dvharmonic index: (9)HG=∑vu∈EG2du+dvinverse sum index:(10)IG=∑vu∈EGdudvdu+dvaugmented Zagreb index:(11)AG=∑vu∈EGdudvdu+dv-23.We refer to [33–45] for detailed survey about the above defined indices and applications. Tables exhibited in [15–19] relate some notable degree-based topological indices with M-polynomial with the following notations [17]:(12)Dx=x∂(fx,y∂x,Dy=y∂(fx,y∂y,Sx=∫0xft,ytdt,Sy=∫0yfx,ttdt,Jfx,y=fx,x,Qαfx,y=xαfx,y.
3. Computational Results
In this section we give our computational results.
3.1. M-Polynomials and Degree-Based Indices for PAMAM Dendrimers
Polyamidoamine (PAMAM) dendrimers are hyperbranched polymers with unparalleled molecular uniformity, narrow molecular weight distribution, defined size and shape characteristics, and a multifunctional terminal surface. These nanoscale polymers consist of an ethylenediamine core, a repetitive branching amidoamine internal structure, and a primary amine terminal surface. Dendrimers are “grown” off a central core in an iterative manufacturing process, with each subsequent step representing a new “generation” of dendrimer. Increasing generations (molecular weight) produce larger molecular diameters, twice the number of reactive surface sites and approximately double the molecular weight of the preceding generation. PAMAM dendrimers also assume a spheroidal, globular shape at generation 4 and above (see molecular simulation below). Their functionality is readily tailored, and their uniformity, size, and highly reactive “molecular Velcro” surfaces are the functional keys to their use. Here we consider PD1, which denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations Gn with n growth stages, and PD2, the PAMAM dendrimers with different core generated by dendrimer generators Gn with n growth stages. DS1 is kinds of PAMAM dendrimers with n growth stages
Theorem 2.
For the PAMAM dendrimers PD1, we have(13)MPD1,x,y=3·2nxy2+32n+1-1xy3+92n+1-1x2y2+37·2n-4x2y3.
Proof.
Let PD1 denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations Gn with n growth stages.
The edge set of PD1 has following four partitions: (14)E1,2=e=uv∈EPD1∣du=1,dv=2,E1,3=e=uv∈EPD1∣du=1,dv=3,E2,2=e=uv∈EPD1∣du=2,dv=2,E2,3=e=uv∈EPD1∣du=2,dv=3.Now(15)E1,2=3·2n,E1,3=6·2n-3,E2,2=18·2n-9,and(16)E2,3=21·2n-12.MPD1;x,y=∑i≤jmijPD1xiyj=∑1≤2m12PD1xy2+∑1≤3m13PD1x1y3+∑2≤2m22PD1x2y2+∑2≤3m23PD1x2y3=∑uv∈E1,2m12PD1xy2+∑uv∈E1,3m13PD1xy3+∑uv∈E2,2m22PD1x2y2+∑uv∈E2,3m23PD1x2y3=E1,2xy2+E1,3x1y3+E2,2x2y2+E2,3x2y3=3×2nxy2+6×2n-3xy3+18×2n-9x2y2+21×2n-12x2y3=3×2nxy2+32n+1-1xy3+92n+1-1x2y2+37×2n-4x2y3.
Theorem 3.
For the PAMAM dendrimers PD1, we have
M1(G)=105×2n+1-108.
M2(G)=111×2n+1-117.
Mm2(G)=2n+1+19×2n-1-21/4.
RαG=3×2n+α+3α+1+22α×92n+1-1+2α×3α+17·2n-4.
RαG=3×2n-α+1/3α-1+9/22α2n+1-1+1/3α-1×2α7·2n-4.
SSD(G)=7×2n+3+3×2n+1+47×2n-54.
HG=7/5×2n+4-54/5
I(G)=497/20×2n+1-513/20.
A(G)=3×2n+3+21×2n+2+369×2n-2-753/8.
Proof.
Let PD1 denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations Gn with n growth stages. Let(17)MG;x,y=fx,y=3×2nxy2+32n+1-1xy3+92n+1-1x2y2+37×2n-4x2y3.Then(18)Dxfx,y=3×2nxy2+32n+1-1xy3+182n+1-1x2y2+67×2n-4x2y3.Dyfx,y=3×2n+1xy2+92n+1-1xy3+182n+1-1x2y2+97×2n-4x2y3.,DyDxfx,y=3×2n+1xy2+92n+1-1xy3+362n+1-1x2y2+187×2n-4x2y3,SxSyfx,y=3×2n-1xy2+2n+1-1xy3+942n+1-1x2y2+127×2n-4x2y3,DxαDyαfx,y=3×2n+αxy2+3α+12n+1-1xy3+22α×92n+1-1x2y2+2α×3α7×2n-4x2y3,SxαSyα(fx,y=3×2n-αxy2+13α-12n+1-1xy3+922α2n+1-1x2y2+13α-1×2α7×2n-4x2y3,SyDxfx,y=3×2n-1xy2+2n+1-1xy3+92n+1-1x2y2+27×2n-4x2y3,SxDyfx,y=3×2n-1xy2+92n+1-1xy3+92n+1-1x2y2+927×2n-4x2y3,SxJfx,y=2nx3+32n+1-1x4+357×2n-4x5,SxJDxDyfx,y=2n+1x3+4542n+1-1x4+1857×2n-4x5,Sx3Q-2JDx3Dy3fx,y=3×2n+3x+36982n+1-1x2+127×2n-4x3.
(1) First Zagreb Index(19)M1G=Dx+Dyfx,yx=y=1=105×2n+1-108.
(2) Second Zagreb Index(20)M2G=DyDxfx,yx=y=1=111×2n+1-117.
(3) Modified Second Zagreb Index(21)Mm2G=SxSyfx,yx=y=1=2n+1+19×2n-1-214.
(8) Inverse Sum Index(26)IG=SxJDxDyfx,yx=1=49720×2n+1-51320.
(9) Augmented Zagreb Index
Theorem 4.
For the PAMAM dendrimers PD2, we have(27)MPD2,x,y=2n+2xy2+42n+1-1xy3+24·2n-11x2y2+142n+1-1x2y3.
Proof.
Let PD2 be the PAMAM dendrimers with different core generated by dendrimer generators Gn with n growth stages. Then the edge set of PD2 has following four partitions: (28)E1,2=e=uv∈EPD2∣du=1,dv=2,E1,3=e=uv∈EPD2∣du=1,dv=3,E2,2=e=uv∈EPD2∣du=2,dv=2,E2,3=e=uv∈EPD2∣du=2,dv=3.Now (29)E1,2=4·2n,E1,3=8·2n-4,E2,2=24·2n-11,and(30)E2,3=28·2n-14.MPD2;x,y=∑i≤jmijPD2xiyj=∑1≤2m12PD2xy2+∑1≤3m13PD2x1y3+∑2≤2m22PD2x2y2+∑2≤3m23PD2x2y3=∑uv∈E1,2m12PD2xy2+∑uv∈E1,3m13PD2xy3+∑uv∈E2,2m22PD2x2y2+∑uv∈E2,3m23PD2x2y3=E1,2xy2+E1,3x1y3+E2,2x2y2+E2,3x2y3=4·2nxy2+8·2n-4xy3+24·2n-11x2y2+28·2n-14x2y3=2n+2xy2+42n+1-1xy3+24·2n-11x2y2+142n+1-1x2y3.
Theorem 5.
For the PAMAM dendrimers PD2, we have
M1(G)=57×2n+3-26.
M2(G)=37×2n+3-140.
Mm2(G)=23/3×2n+1-77/12.
RαG=2n+α+2+4×3α+2α+1×3α×72n+1-1+22α24·2n-11.
RαG=2n+4/3α+14/6α2n+1-1+1/22α24·2n-11.
SSD(G)=109/3×2n+2-197/3.
HG=7/5×2n+6-131/10.
I(G)=497/15×2n+1-154/5.
A(G)=475×2n-427/2.
Theorem 6.
For the PAMAM dendrimers DS1, we have(31)MDS1;x,y=4·3nxy4+103n-1x2y2+43n-1x2y4.
Proof.
Let DS1 be kinds of PAMAM dendrimers with n growth stages.
The edge set of DS1 has the following three partitions: (32)E1,4=e=uv∈EDS1∣du=1,dv=4,E2,2=e=uv∈EDS1∣du=2,dv=2,E2,4=e=uv∈EDS1∣du=2,dv=4.Now(33)E1,4=4·3n,E2,2=10·3n-10,and(34)E2,4=4·3n-4.MDS1;x,y=∑i≤jmijDS1xiyj=∑1≤4m14DS1xy4+∑2≤2m22DS1x2y2+∑2≤4m24DS1x2y4=∑uv∈E1,4m14DS1xy4+∑uv∈E2,2m22DS1x2y2+∑uv∈E2,4m24DS1x2y4=E1,4xy4+E2,2x2y2+E2,4x2y4=4·3nxy4+10·3n-10x2y2+4·3n-4x2y4=4·3nxy4+103n-1x2y2+43n-1x2y4.
Theorem 7.
For the PAMAM dendrimers DS1, we have
M1(G)=47×3n+1-16.
M2(G)=811×3n-9.
Mm2(G)=4×3n-3.
RαG=3n×4α+1+2α+1×5+23α+23n-1.
RαG=3n/4α-1+5/22α-1+1/23α-23n-1.
SSD(G)=47×3n-30
HG=119/15×3n-19/5
I(G)=278/15×3n-46/3.
A(G)=3280/27×3n-112.
3.2. M-Polynomials and Degree-Based Indices for Polyomino Chains
From the geometric point of view, a polyomino system is a finite 2-connected plane graph in which each interior cell is encircled by a regular square. In other words, it is an edge-connected union of cells in the planar square lattice. Polyomino chain is a particular polyomino system such that the joining of the centers (set ci as the center of the ith square) of its adjacent regular composes a path c1,c2,c3,…cn.
Let Bn be the set of polyomino chains with n squares. There are 2n+1 edges in every Bn∈Bn, where Bn is named as a linear chain and denoted by Ln if the subgraph of Bn induced by the vertices with d(v)=3 is a molecular graph with exactly n-2 squares. Also, Bn can be called a zigzag chain and labelled as Zn if the subgraph of Bn is induced by the vertices with d(v)>2 is Pn.
The angularly connected, or branched, squares constitute a link of a polyomino chain. A maximal linear chain (containing the terminal squares and kinks at its end) in the polyomino chains is called a segment of polyomino chain. Let l(S) be the length of S which is calculated by the number of squares in S. For any segment S of a polyomino chain, we get l(S)∈2,3,4,…,n. Furthermore, we deduce l1=n and m=1 for a linear chain Ln with n squares and li=2 and m=n-1 for a zigzag chain Zn with n squares.
In what follows, we always assume that a polyomino chain consists of a sequence of segments S1,S2,S3,…Sn and L(Si)=li, where m≥1 and i∈2,3,4,…,m. We derive that ∑i=1mli=n+m-1.
Theorem 8.
For a linear polyomino chain Ln, we have MLn;x,y=2x2y2+4x2y3+3n-5x3y3.
Proof.
Let Ln be the polyomino chain with n squares where l1=n and m=1. Ln is called the linear chain.
The edge set of Ln has the following three partitions: (35)E2,2=e=uv∈ELn∣du=2,dv=2,E2,3=e=uv∈ELn∣du=2,dv=3,E3,3=e=uv∈ELn∣du=3,dv=3.Now(36)E2,2=2,and(37)E2,3=4,E3,3=3n-5.(38)MLn;x,y=∑i≤jmijLnxiyj=∑2≤2m22Lnx2y2+∑2≤3m23Lnx2y3+∑3≤3m33Lnx3y3=∑uv∈E2,2m22Lnx2y2+∑uv∈E2,3m23Lnx2y3+∑uv∈E3,3m33Lnx3y3=E2,2x2y2+E2,3x2y3+E3,3x3y3=2x2y2+4x2y3+3n-5x3y3.
Theorem 9.
For a linear polyomino chain Ln, we have the following:
M1(G)=18n-2.
M2(G)=27n-13.
Mm2(G)=1/3n-11/18.
RαG=22α+1+2α+2·3α+32α3n-5.
RαG=1/2α-1+22-α/3α+1/32α3n-5.
SSD(G)=6n+8/3.
HG=n+14/15.
I(G)=9/2n-7/10.
A(G)=2187/64n-573/64.
Theorem 10.
Let Zn be zigzag polyomino chain with n squares such that li=2 and m=n-1. Then(39)MZn,x,y=2x2y2+4x2y3+2m-1x2y4+2x3y4+3n-2m-5x4y4.
Proof.
Let Zn be zigzag polyomino chain with n squares such that li=2 and m=n-1. Polyomino chain consists of a sequence of segments S1,S2,…Sm and lSi=li where m≥1 and i∈1,2,…,m.
The edge set of Zn has the following five partitions:(40)E2,2=e=uv∈EZn∣du=2,dv=2,E2,3=e=uv∈EZn∣du=2,dv=3,E2,4=e=uv∈EZn∣du=2,dv=4,E3,4=e=uv∈EZn∣du=3,dv=4,E4,4=e=uv∈EZn∣du=4,dv=4.Now(41)E2,2=2,E2,3=4,E2,4=2m-1,E3,4=2,and(42)E4,4=3n-2m-5.MZn;x,y=∑i≤jmijZnxiyj=∑2≤2m22Znx2y2+∑2≤3m23Znx2y3+∑2≤4m24Znx2y4+∑3≤4m34Znx3y4+∑4≤4m44Znx4y4=∑uv∈E2,2m22Znx2y2+∑uv∈E2,3m23Znx2y3+∑uv∈E2,4m24Znx2y4+∑uv∈E3,4m34Znx3y4+∑uv∈E4,4m44Znx4y4=E2,2x2y2+E2,3x2y3+E2,4x2y4+E3,4x3y4+E4,4x4y4=2x2y2+4x2y3+2m-1x2y4+2x3y4+3n-2m-5x4y4.
Theorem 11.
For the Zigzag polyomino chain Zn for n≥2, we have the following:
For the polyomino chain with n squares and of m segments S1 and S2 satisfying l1=2 and l2=n-1, Bn1n≥3, we have(43)MBn1,x,y=2x2y2+5x2y3+x2y4+3n-10x3y3+3x3y4.
Proof.
Let Bn1n≥3 be the polyomino chain with n squares and of m segments S1 and S2 satisfying l1=2 and l2=n-1. The edge set of Bn1n≥3 has the following five partitions: (44)E2,2=e=uv∈EBn1∣du=2,dv=2,E2,3=e=uv∈EBn1∣du=2,dv=3,E2,4=e=uv∈EBn1∣du=2,dv=4,E3,3=e=uv∈EBn1∣du=3,dv=3,E3,4=e=uv∈EBn1∣du=3,dv=4.Now(45)E2,2=2,E2,3=5,E2,4=1,E3,3=3n-10,and(46)E3,4=3.(47)MBn1;x,y=∑i≤jmijBn1xiyj=∑2≤2m22Bn1x2y2+∑2≤3m23Bn1x2y3+∑2≤4m24Bn1x2y4+∑3≤3m33Bn1x3y3+∑3≤4m34Bn1x3y4=∑uv∈E2,2m22Bn1x2y2+∑uv∈E2,3m23Bn1x2y3+∑uv∈E2,4m24Bn1x2y4+∑uv∈E3,3m33Bn1x3y3+∑uv∈E3,4m34Bn1x3y4=E2,2x2y2+E2,3x2y3+E2,4x2y4+E3,3x3y3+E3,4x3y4=2x2y2+5x2y3+x2y4+3n-10x3y3+3x3y4.
Theorem 13.
For the polyomino chain with n squares and of m segments S1 and S2 satisfying l1=2 and l2=n-1, Bn1n≥3, we have the following:
M1(G)=18n.
M2(G)=27n-8.
Mm2(G)=n/3+43/72.
RαG=22α+1+6α×5+23α+32α3n-10+3α+1×4α.
RαG=1/22α-1+5/6α+1/23α+1/32α3n-10+1/22α×3α-1.
SSD(G)=6n+43/12.
HG=n+6/7.
I(G)=27n-709/10.
A(G)=2187/64n-33737/4000.
Theorem 14.
For polyomino chain with n squares and m segments S1,S2,…,Smm≥3 satisfying l1=lm=2 and l2,…,lm-1≥3, Bn2n≥4, we have(48)MBn2,x,y=2x2y2+2mx2y3+2x2y4+3n-2m+1x3y3+22m-3x3y4.
Proof.
Let Bn2n≥4 be a polyomino chain with n squares and m segments S1,S2,…,Smm≥3 satisfying l1=lm=2 and l2,…,lm-1≥3. Then the edge set of Bn2n≥4 has the following five partitions: (49)E2,2=e=uv∈EBn2∣du=2,dv=2,E2,3=ne=uv∈EBn2∣du=2,dv=3,E2,4=e=uv∈EBn2∣du=2,dv=4,E3,3=e=uv∈EBn2∣du=3,dv=3,E3,4=e=uv∈EBn2∣du=3,dv=4.Now(50)E2,2=2,E2,3=2m,E2,4=2,E3,3=3n-6m+3,and(51)E4,4=4m-6.(52)MBn2;x,y=∑i≤jmijBn2xiyj=∑2≤2m22Bn2x2y2+∑2≤3m23Bn2x2y3+∑2≤4m24Bn2x2y4+∑3≤3m33Bn2x3y3+∑3≤4m34Bn2x3y4=∑uv∈E2,2m22Bn2x2y2+∑uv∈E2,3m23Bn2x2y3+∑uv∈E2,4m24Bn2x2y4∑uv∈E3,3m33Bn2x3y3+∑uv∈E3,4m34Bn2x3y4=E2,2x2y2+E2,3x2y3+E2,4x2y4+E3,3x3y3+E3,4x3y4=2x2y2+2mx2y3+2x2y4+3n-6m+3x3y3+4m-6x3y4=2x2y2+2mx2y3+2x2y4+3n-2m+1x3y3+22m-3x3y4
Theorem 15.
For polyomino chain with n squares and m segments S1,S2,…,Smm≥3 satisfying l1=lm=2 and l2,…,lm-1≥3, Bn2n≥4, we have the following:
Topological indices calculated in this paper help us to guess biological activities, chemical reactivity, and physical features of under-study dendrimers and polyomino chains. For example, Randić index is useful for determining physiochemical properties of alkanes as noticed by chemist Milan Randić in 1975. He noticed the correlation between the Randić index and several physicochemical properties of alkanes like boiling point, vapor pressure, enthalpies of formation, surface area, and chromatographic retention times. Hence our results are helpful in determination of the significance of PAMAM dendrimers and polyomino chains in pharmacy and industry.
Data Availability
All data required for this research is included within this paper.
Conflicts of Interest
The authors do not have any conflicts of interest.
Authors’ Contributions
All authors contributed equally to this paper.
Acknowledgments
The first author is funded by Dong-A University, Korea.
GoT.GoodmanJ. E.O'RourkeJ.1997CRC Press, Boca Raton, FLCRC Press Series on Discrete Mathematics and its ApplicationsMR1730156GolombS. W.1996Princeton University PressAlonsoL.CerfR.The three dimensional polyominoes of minimal area199631 R1392-s2.0-0041543083Zbl0885.05056AstrucD.BoisselierE.OrnelasC.Dendrimers designed for functions: from physical, photophysical, and supramolecular properties to applications in sensing, catalysis, molecular electronics, photonics, and nanomedicine201011041857195910.1021/cr900327d2-s2.0-77951234285VögtleF.RichardtG.WernerN.2009FarahaniM. R.GaoW.KannaM. R.The connective eccentric index for an infinite family of dendrimers20155S4766771GaoW.ShiL.FarahaniM. R.Distance-based indices for some families of dendrimer nanostars20164621681862-s2.0-84969988831GaoW.WangW.Degree-based indices of polyhex nanotubes and dendrimer nanostar2016133157715832-s2.0-8497727367310.1166/jctn.2016.5083GaoW.FarahaniM. R.The hyper-zagreb index for an infinite family of nanostar dendrimer20172025155232-s2.0-8502049413210.1080/09720529.2016.1220088GaoW.FarahaniM. R.ShiL.The forgotten topological index of some drug structures20163215795852-s2.0-85015246127GaoW.WangY.WangW.ShiL.The first multiplication atom-bond connectivity index of molecular structures in drugs20172545485552-s2.0-8501834092110.1016/j.jsps.2017.04.021GaoW.YounasM.FarooqA.M-Polynomials and Degree-Based Topological Indices of the Crystallographic Structure of Molecules201884107ZhangH.ZhangF.The Clar covering polynomial of hexagonal systems1996691-214716710.1016/0166-218X(95)00081-2MR1400510WienerH.Structural determination of paraffin boiling points194769117202-s2.0-854425410710.1021/ja01193a005DeutschE.KlavžarS.M-Polynomial and degree-based topological indices2015693102MunirM.NazeerW.RafiqueS.KangS. M.M-polynomial and related topological indices of nanostar dendrimers2016899710.3390/sym8090097MR3554576MunirM.NazeerW.NizamiA. R.RafiqueS.KangS. M.M-polynomials and topological indices of titania nanotubes201681111710.3390/sym8110117MR3576199KwunY. C.MunirM.NazeerW.RafiqueS.Min KangS.M-Polynomials and topological indices of V-Phenylenic Nanotubes and Nanotori20177110.1038/s41598-017-08309-yMunirM.NazeerW.RafiqueS.NizamiA. R.KangS. M.Some computational aspects of boron triangular nanotubes20179610.3390/sym9010006MR3618930MunirM.NazeerW.RafiqueS.KangS. M.M-polynomial and degree-based topological indices of polyhex nanotubes201681214910.3390/sym8120149MR3597307DobryninA. A.EntringerR.GutmanI.Wiener index of trees: theory and applications200166321124910.1023/a:1010767517079MR1843259Zbl0982.050442-s2.0-0035329265MoharB.PisanskiT.How to compute the Wiener index of a graph1988232672772-s2.0-000002468610.1007/BF01167206ZhouB.GutmanI.Relations between Wiener, hyper-Wiener and Zagreb indices20043941–3939510.1016/j.cplett.2004.06.1172-s2.0-3843121931RandićM.On characterization of molecular branching197597236609661510.1021/ja00856a0012-s2.0-8644280181KangS. M.ZahidM. A.NazeerW.GaoW.Calculating the Degree-based Topological Indices of Dendrimers2018161681688BollobásB.ErdösP.Graphs of extremal weights199850225233MR1670561Zbl0963.050682-s2.0-0032411939AmićD.BešloD.LučićB.NikolićS.TrinajstićN.The Vertex-Connectivity Index Revisited199838581982210.1021/ci980039bHuY.LiX.ShiY.XuT.GutmanI.On molecular graphs with smallest and greatest zeroth-order general randić index2005542425434MR2201595CaporossiG.GutmanI.HansenP.PavlovićL.Graphs with maximum connectivity index200327185902-s2.0-034570220910.1016/S0097-8485(02)00016-5GutmanI.DasK. C.The first Zagreb index 30 years after20045018392DasK. C.GutmanI.Some properties of the second Zagreb index200452113GutmanI.An exceptional property of first Zagreb index2014723733740MR3288983Zbl06704640HuangY.LiuB.GanL.Augmented Zagreb index of connected graphs2012672483494MR2951681Zbl1289.05065NazeerW.FarooqA.YounasM.MunirM.KangS.On Molecular Descriptors of Carbon Nanocones20188392KwunY.VirkA.NazeerW.RehmanM.KangS.On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I [p, q] and Si2C3-II [p, q]2018108320KangS.IqbalZ.IshaqM.SarfrazR.AslamA.NazeerW.On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers201810723710.3390/sym10070237KwunY. C.AliA.NazeerW.Ahmad ChaudharyM.KangS. M.M-Polynomials and Degree-Based Topological Indices of Triangular, Hourglass, and Jagged-Rectangle Benzenoid Systems201820188821395010.1155/2018/8213950LiuJ.-B.PanX.-F.YuL.LiD.Complete characterization of bicyclic graphs with minimal Kirchhoff index2016200951072-s2.0-8495932126410.1016/j.dam.2015.07.001LiuJ.-B.WangW.-R.ZhangY.-M.PanX.-F.On degree resistance distance of cacti201620321722510.1016/j.dam.2015.09.006MR3474729Zbl1332.050482-s2.0-84951093568LiuJ.-B.PanX.-F.Minimizing Kirchhoff index among graphs with a given vertex bipartiteness201629184882-s2.0-8497761959510.1016/j.amc.2016.06.017GaoW.JamilM. K.JavedA.FarahaniM. R.WangS.LiuJ.-B.Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs201720175607945010.1155/2017/60794502-s2.0-85013308914LiuJ.-B.GaoW.SiddiquiM. K.FarahaniM. R.Computing three topological indices for Titania nanotubes TiO2[m,n]20161332552602-s2.0-8499566612710.1016/j.akcej.2016.07.001LiuQ.LiuJ.-B.CaoJ.The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs20161774414462-s2.0-8495929110410.1016/j.neucom.2015.11.060LiuJ.-B.WangS.WangC.HayatS.Further results on computation of topological indices of certain networks20171113206520712-s2.0-8501695775410.1049/iet-cta.2016.1237HuoY.LiuJ.-B.BaigA. Q.SajjadW.FarahaniM. R.Connective eccentric index of NAnm manotube2017144183218362-s2.0-8502128725110.1166/jctn.2017.6512