Pharmacological Characteristics Analysis of Two Molecular Structures

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Abstract

Each year a large number of new diseases were found worldwide, which requires the development of new drugs to cure these diseases. In this process, researchers need to do a lot of work to test the effectiveness of new drugs and side effects. Due to the intrinsic connection between the characteristics of compound and its molecular structure, methods of pharmaceutical theory are widely used in the analysis of the features of the drug. By calculating the chemical indices of drug molecular structure, scientists could learn the chemistry and pharmacy characteristics of the corresponding drugs. In this paper, from the theoretical perspective, we state the following conclusions: (1) the exact expression of generalized degree distance for starlike tree is determined; (2) the eccentricity related indices of hetrofunctional dendrimer are discussed. The results obtained have broad application prospects in the pharmaceutical sciences.

1 Introduction

It’s revealed by drug testing from the early research that the physico-chemical and pharmacological properties of drugs are closely related to their molecular structures. It raised much attention from the theoretical researchers, and until now, many topological indices are defined as useful as numerical parameters of drug structures which play an important role on understanding the properties of drugs.

In theoretical pharmacy model, a molecular structure of each drug is denoted as a molecular graph G (each atom is expressed as a vertex and each chemical bound is represented as an edge), then a topological index can be regarded as a score function f : G → ℝ+ which maps each molecular graph to a read positive number. In the past four decades, many indices are introduced by scholars from the engineering application prospects, such as Zagreb index, Wiener index, sum connectivity index, Gutman index and harmonic index which reflect several structural characteristics of molecules and drugs. There were many contributions to report these degree-based and distance-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao et al. [37] and Gao and Wang [810] for more details). The notation and terminology that were used but undefined in this paper can be found in [11].

Now, we present some important indices which will be computed in the next section. The Shultz polynomial is denoted as

Sc(G,x)=Σ{u,v}V(G)(d(u)+d(v))xd(u,v).

The additively weighted Harary index (also called, reciprocal degree distance) is defined as

HA(G)=Σ{u,v}V(G)d(u)+d(v)d(u,v).

As the extension, the generalized degree distance is denoted as

Hλ(G)=Σ{u,v}V(G)(d(u)+d(v))dλ(u,v).

And, the corresponding polynomial can be stated as

Hλ(G,x)=Σ{u,v}V(G)(d(u)+d(v))dλ(u,v).

Some results on above indices and polynomials can refer to Alizadeh et al. [12], Sedlar [13], Pourfaraj and Ghorbani [14], Pattabiraman and Vijayaragavan [15], and Hamzeh et al. [16] and [17].

Let d(v) be the degree of vertex v, and ec(v) be the eccentricity of vertex v which is denoted as the largest distance between v and any other vertex in molecular graph G. The first atom-bond connectivity index (ABC index) is defined by Estrada et al. [18] which is stated as

ABC(G)=ΣuvE(G)d(u)+d(v)2d(u)d(v).

Then, the eccentricity version atom bond connectivity index (called the fifth ABC index) is denoted as

ABC5(G)=ΣuvE(G)ec(u)+ec(v)2ec(u)ec(v).

The first geometric-arithmetic index (GA index) introduced by Vukičević and Furtula [19] which can be formulated as

GA(G)=ΣuvE(G)2d(u)d(v)d(u)+d(v).

Several contributions on geometric-arithmetic index can be found in Zhou et al. [20], Rodríguez and Sigarreta [21], [22] and [23], Husin et al. [24], Bahrami and Alaeiyan [?], Sigarreta [26], Divnic et al. [27], Das et al. [28], Mahmiani et al. [29], Fath-Tabar et al. [30] and [31], Das et al. [32], Gutman and Furtula [33], Furtula and Gutman [34], and Shabani et al. [35]. The the eccentricity version geometric-arithmetic index (called the fourth GA index) is defined by Lee et al. [36] which can be expressed as

GA4(G)=ΣuvE(G)2ec(u)ec(v)ec(u)+ec(v).

Furthermore, the first and the second multiplicative version of eccentricity index are described as

Π1*(G)=ΠuvE(G)(ec(u)+ec(v))

and

Π2*(G)=ΠuvE(G)(ec(u)+ec(v)),

respectively. Moreover, as related polynomials, the fourth and sixth Zagreb polynomials are defined as

Zg4(G,x)=ΣuvE(G)xec(u)+ec(v)

and

Zg6(G,x)=ΣuvE(G)xec(u)ec(v),

respectively.

Although there have been several works in topological indices of material structures, the research on topological indices for certain special drug structures is still largely limited. Furthermore, as a widespread and critical drug structure, dendrimer and starlike molecular graphs are widely used in medical science and frequently appears in new drug structures (see Kobeissi and Mollard [37], Omidi and Tajbakhsh [38], Omidi and Vatandoost [39], Betancur et al. [40] and Farooq et al. [41] for more details). It inspires us to obtain the exact expressions of some special topological indices for important classes of starlike molecular graphs and dendrimer.

The main aim of this paper is to study the generalized degree distance of starlike tree and eccentricity related indices of hetrofunctional dendrimer.

2 Main results and proofs

The purpose of this section is to present our main results and detail proofs.

2.1 Indices study of starlike tree

Let D(G) = (d1, d2, ··· , dn) be the degree sequence of the graph G with d1d2 ··· dn, where di denotes the degree of the i-th vertex in G. Moreover, D(G)=(d1a1,d2a2,,dtat) implies that G has ai vertices with degree di for i ∊ {1, 2, ··· ,t}.

A double star Sp,q (the detailed structure can refer to Figure 1(a)) is a tree which is yielded from K1,p and K1,q−1 by identifying a pendent vertex of K1,p with the center of K1,q−1, where 1 < pq. Hence, for a double star Sp,q with order n, we get p + q = n and pn2. Furthermore, if p=n2 and q=n2, then Sp,q is called a balanced double star.

Fig. 1
Fig. 1

Two classes of starlike tree

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00008

Let (c1, c2, ··· , cd) be a partition of order n. The starlike tree can be constructed using the following method:

(i) Let S1, S2, ··· , Sd be the stars and v1, v2, ··· ,vd be their center vertices. Set |E(S1)| = c1 1, |E(S2)| = c2 1, ··· ,|E(Sd)| = cd 1;

(ii) Add a vertex v0 and then adjacent to the center vertices v1, v2, ··· ,vd of S1, S2, ··· ,Sd respectively.

In this way, we can deduce a tree T with diameter at most 4, and dT (v1) = c1, dT (v2) = c2, ···, dT (vd) = cd respectively. Moreover, we infer |V (T )| = n + 1, E(T)=Σi=1dci=n. We denote this molecular structure as S(c1, c2, ··· ,cd) shown in Figure 1(b). Now, our main result in this part is stated as follows.

Theorem 1.

Let S(c1, c2, ··· ,cd) be a starlike tree described above. Then, we have

Hλ(S(c1,c2,,cd))=Σi=1d(ci2+ci)+d2d+(Σi=1d(2(ci12)+(d1)ci)+(d+1)(nd))2λ+(Σi=1d(ci+1)Σi=1d(ci1)Σi=1d(ci21))3λ+2((nd2)Σi=1d(ci12))4λ.

Proof.

For any pairs of vertices (x, y) ⊆ V (S(c1, c2, ··· ,cd)), we get d(x, y) ≤ 4. Thus, we have d(x, y) = k where k ∊ {1, 2, 3, 4}. The vertex set of S(c1, c2, ··· ,cd) can be divided into three classes: (1) the center v0; (2) v1, ··· , vd; (3) the leaves w1, w2, ··· , wn−d. The following discussion can be divided into four parts.

  • Since |E(S(c1, c2, ··· ,cd))| = n, there are n pairs with d(x, y) = 1, and the total contribution of this part to generalized degree distance is
    Σi=1d(ci+1)(ci1)+Σi=1d(ci+d)=Σi=1d(ci2+ci)+d2d.
  • Vertices x and y which in pairs (x, y) = (v0, wi), (vi, vj) or (wi, wj) satisfy d(x, y) = 2. The total contribution of this part to generalized degree distance is
    (Σi=1d(2(ci12)+(d1)ci)+(d+1)(nd))2λ.
  • We get d(vi, wj) = 3 where i ≠ = j, and the total contribution of this part to generalized degree distance is
    (Σi=1d(ci+1)Σi=1d(ci1)Σi=1d(ci21))3λ.
  • We get d(wi, wj) = 4 if and only if wi and wj are not the neighbors of the same vertex vk (1 ≤ kd). The total contribution of this part to generalized degree distance is
    2((nd2)Σi=1d(ci12))4λ.

By summing up the above results, we get the desired result.

In view of Theorem 1, we deduce the following corollaries.

Corollary 1.

The Shultz polynomial of starlike tree S(c1, c2, ··· , cd) is

Sc(c1,c2,,cd),x)=(Σi=1d(ci2+ci)+d2d)x+(Σi=1d(2(ci12)+(d1)ci)+(d+1)(nd))x2+(Σi=1d(ci+1)Σi=1d(c11)Σi=1d(ci21))x3+2((nd2)Σi=1d(ci12))x4.

Corollary 2.

The additively weighted Harary index (reciprocal degree distance) of starlike tree S(c1, c2, ··· , cd) is

HA(S(c1,c2,,cd))=(Σi=1d(ci2+ci)+d2d)+Σi=1d(ci12)+(d1)ci)+(d+1)(nd)+(nd2)2+Σi=1d(ci+1)Σi=1d(ci1)Σi=1d(ci21)3.

Corollary 3.

The generalized degree distance polynomial of starlike tree S(c1, c2, ··· ,cd) is

Hλ(S(c1,c2,,cd),x)=(Σi=1d(ci2+ci)+d2d)x+(Σi=1d(2(ci12)+(d1)ci)+(d+1)(nd))x2λ+(Σi=1d(ci+1)Σi=1d(ci1)Σi=1d(ci21))x3λ+2((nd2)Σi1d(ci12))x4λ.

2.2 Eccentricity related indices of hetrofunctional dendrimer

As macromolecules, hetrofunctional dendrimers D[n] (here n is denoted as the total stage number) have been widely used in pharmacy and medicine. As an example, the structure of D[6] is presented in Figure 2, and the different growth stages are depicted in Figure 3 and Figure 4. Obviously, we have |V(D[n])|=|E(D[n])|={40×2t38,if n=2t24×2t+138,if n=2t+1, where t is a positive integer.

Fig. 2
Fig. 2

The structure of D[6]

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00008

Fig. 3
Fig. 3

The core of D[1] and D[2]

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00008

Fig. 4
Fig. 4

One branch of D[3], D[4] and D[5]

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00008

Our proof follows the technology in Farooq et al. [41], and the whole results can be divided into two parts according to the parity of n.

Theorem 2.

If n = 1, then

GA4(D[1])=2+1659+43011+44213,ABC5(D[1])=32+275+2310+21142,Π1*(D[1])=8294112132,Π2*(D[1])=162204302422,Zg4(D[1])=2x8+4x9+2x11+2x13,Zg6(D[1])=2x16+4x20+2x30+2x42.

If n = 2, then

GA4(D[2])=2+163323+83925+418227+821029+321531+641733+243435+243837+169539+1610541,

ABC5(D[2])=45121+4744+223156+225182+4970+2915+23117+433334+433538+23795+239105,

Π1*(D[2])=222234252272294314338354374394414,Π2*(D[2])=12121324156218222104240427283064342438044204,Zg4(D[2])=2x22+4x23+2x25+2x27+4x29+4x31+8x33+4x35+4x37+4x39+4x41,

Zg6(D[2])=2x121+4x132+2x156+2x182+4x210+4x240+8x272+4x306+4x342+4x380+4x420.

If n = 3, then

GA4(D[3])=2+321530+161733+123435+243837+169539+1610541+846243+850645+1613247+80649+402651+247853+482155+1620357,

ABC5(D[3])=4157+2915+123117+211102+433538+23795+21335+441462+443506+21346+25476+454926+431726+235321+255203,

Π1*(D[3])=302314333352374394414434454474494514534554574,

Π2*(D[3])=225224042722306234243804420446245064552460046504702475648124,

Zg4(D[3])=2x30+4x31+2x33+2x35+4x37+4x39+4x41+4x43+4x45+4x47+4x49+4x51+4x53+4x55+4x57,

Zg6(D[3])=2x225+4x240+2x272+2x306+4x342+4x420+4x462+4x506+4x552+4x600+4x650+4x702+4x756+4x812.

If n ≥ 4 and n ≡ 0(mod2), then

GA4(D[n])=2+8(11t)(11t+1)22t+1+4(11t+1)(11t+2)22t+3+4(11t+2)(11t+3)22t+5+8(11t+3)(11t+4)22t+7+i=1t1(2i+2(11t+11i7)(11t+11i6)22t+22i13+2i+3(11t+11i6)(11t+11i5)22t+22i11+2i+2(11t+11i5)(11t+11i4)22t+22i9+2i+2(11t+11i4)(11t+11i3)22t+22i7+2i+2(11t+11i3)(11t+11i2)22t+22i5+2i+2(11t+11i2)(11t+11i1)22t+22i3+2i+2(11t+11i1)(11t+11i)22t+22i1+2i+2(11t+11i)(11t+11i+1)22t+22i+1+2i+2(11t+11i+1)(11t+11i+2)22t+22i+3+2i+2(11t+11i+2)(11t+11i+3)22t+22i+5+2i+2(11t+11i+3)(11t+11i+4)22t+22i+7)+2t+2(22t7)(22t6)44t13+2t+3(22t6)(22t5)44t11+2t+2(22t5)(22t4)44t9+2t+2(22t4)(22t3)44t7+2t+2(22t3)(22t2)44t5+2t+2(22t2)(22t1)44t3,

ABC5(D[n])=211t22t2+422t1(11t)(11t+1)+222t+1(11t+1)(11t+2)+222t+3(11t+2)(11t+3)+422t+5(11t+3)(11t+4)+Σi=1t1(2i+122t+22i15(11t+11i7)(11t+11i6)+2i+222t+22i13(11t+11i6)(11t+11i5)+2i+122t+22i11(11t+11i5)(11t+11i4)+2i+122t+22i9(11t+11i4)(11t+11i3)+2i+122t+22i7(11t+11i3)(11t+11i2)+2i+122t+22i5(11t+11i2)(11t+11i1)+2i+122t+22i3(11t+11i1)(11t+11i)+2i+122t+22i1(11t+11i)(11t+11i+1)+2i+122t+22i+1(11t+11i+1)(11t+11i+2)+2i+122t+22i+3(11t+11i+2)(11t+11i+3)+2i+122t+22i+5(11t+11i+3)(11t+11i+4)+2t+144t15(22t7)(22t6)+2t+244t13(22t6)(22t5)+2t+144t11(22t5)(22t4)+2i+144t9(22t4)(22t3)+2t+144t7(22t3)(22t2)+2t+144t5(22t2)(22t1),

Π1*(D[n])=(22t)2(22t+1)4(22+3)2(22t+5)2(22+7)4Πi=1t1((22t+22i13)2i+1(22t+22i11)2i+2(22t+22i9)2i+1(22t+22i7)2i+1(22t+22i5)2i+1(22t+22i3)2i+1(22t+22i1)2i+1(22t+22i1)2i+1(22t+22i3)2i+1(22t+22i3)2i+1(22t+22i1)2i+1(22t+22i1)2i+1(22t+22i3)2i+1(22t+22i5)2i+1(22t+22i7)2i+1(44t13)2t+1(44t11)2t+2(44t9)2t+1(44t7)2t+1(44t5)2t+1(44t3)2t+1,

Π2*(D[n])=(11t)8(11t+1)6(11+2)4(11t+3)6(11t+4)4Πi=1t1((11t+11i7)2i+1(11t+11i6)32i+1(11t+11i5)32i+1(11t+11i4)2i+2(11t+11i3)2i+2(11t+11i2)2i+2(11t+11i1)2i+2(11t+11i)2i+2(11t+11i+1)2i+2(11t+11i2)2i+2(11t+11i+3)2i+2(11t+11i4)2i+1(22t7)2i+1(22t6)3.2i+1(22t5)3.2i+1(22t4)2i+2(22t3)2i+2(22t2)2i+2(22t1)2i+1,

Zg4(D[n])=2x22t+4x22t+1+2x22t+3+2x22t+5+4x22t+7+Σi=1t1(2i+1x22t+22i13)+2i+2x22t+22i11+2i+1x22t+22i9+2i+1x22t+22i7+2i+1x22t+22i5+2i+1x22t+22i3+2i+1x22t+22i1+2i+1x22t+22i+1+2i+1x22t+22i+3+2i+1x22t+22i+5+2i+1x22t+22i+7)+2t+1x44t13+2t+2x44t11+2t+1x44t9+2t+1x44t72t+1x44t5+2t+1x44t3,

Zg6(D[n])=2x(11t)(11t)+4x(11t)(11t+1)+2x(11t+1)(11t+2)+2x(11t+2)(11t+3)+4x(11t+3)(11t+4)+i=1t1(2i+1x(11t+11i7)(11t+11i6)+2i+2x(11t+11i6)(11t+11i5)+2i+1x(11t+11i5)(11t+11i4)+2i+1x(11t+11i4)(11t+11i3)+2i+1x(11t+11i3)(11t+11i2)+2i+1x(11t+11i2)(11t+11i1)+2i+1x(11t+11i1)(11t+11i)+2i+1x(11t+11i)(11t+11i+1)+2i+1x(11t+11i+1)(11t+11i+2)+2i+1x(11t+11i+2)(11t+11i+3)+2i+1x(11t+11i+3)(11t+11i+4))+2t+1x(22t7)(22t6)+2t+2x(22t6)(22t5)+2t+1x(22t5)(22t4)+2t+1x(22t4)(22t3)+2t+1x(22t3)(22t2)+2t+1x(22t2)(22t1).

If n ≥ 5 and n ≡ 1(mod2), then

GA4(D[n])=2+8(11t+4)(11t+5)22t+9+4(11t+5)(11t+6)22t+11+4(11t+6)(11t+7)22t+13+8(11t+7)(11t+8)22t+15+8(11t+8)(11t+9)22t+17+16(11t+9)(11t+10)22t+19+8(11t+10)(11t+11)22t+21+8(11t+11)(11t+12)22t+23+8(11t+12)(11t+13)22t+25+8(11t+13)(11t+14)22t+27+8(11t+14)(11t+15)22t+29+8(11t+15)(11t+16)22t+31+8(11t+16)(11t+17)22t+33+8(11t+17)(11t+18)22t+35+i=1t1(2i+3(11t+11i+7)(11t+11i+8)22t+22i+15+2i+3(11t+11i+8)(11t+11i+9)22t+22i+17+2i+4(11t+11i+9)(11t+11i+10)22t+22i+19+2i+3(11t+11i+10)(11t+11i+11)22t+22i+21+2i+3(11t+11i+11)(11t+11i+12)22t+22i+23+2i+3(11t+11i+12)(11t+11i+13)22t+22i+25+2i+3(11t+11i+13)(11t+11i+14)22t+22i+27+2i+3(11t+11i+14)(11t+11i+15)22t+22i+29+2i+3(11t+11i+15)(11t+11i+16)22t+22i+31+2i+3(11t+11i+16)(11t+11i+17)22t+22i+33+2i+3(11t+11i+17)(11t+11i+18)22t+22i+35),

ABC5(D[n])=211t+422t+6+422t+7(11t+4)(11t+5)+222t+9(11t+5)(11t+6)+222t+11(11t+6)(11t+7)+422t+13(11t+7)(11t+8)+422t+15(11t+8)(11t+9)+822t+17(11t+9)(11t+10)+422t+19(11t+10)(11t+11)+422t+21(11t+11)(11t+12)+422t+23(11t+12)(11t+13)+422t+25(11t+13)(11t+14)+422t+27(11t+14)(11t+15)+422t+29(11t+15)(11t+16)+422t+31(11t+16)(11t+17)+422t+33(11t+17)(11t+18)+i=1t1(2i+222t+22i+13(11t+11i+7)(11t+11i+8)+2i+222t+22i+15(11t+11i+8)(11t+11i+9)+2i+322t+22i+17(11t+11i+9)(11t+11i+10)+2i+222t+22i+19(11t+11i+10)(11t+11i+11)+2i+222t+22i+21(11t+11i+11)(11t+11i+12)+2i+222t+22i+23(11t+11i+12)(11t+11i+13)+2i+222t+22i+25(11t+11i+13)(11t+11i+14)+2i+222t+22i+27(11t+11i+14)(11t+11i+15)+2i+222t+22i+29(11t+11i+15)(11t+11i+16)+2i+222t+22i+31(11t+11i+16)(11t+11i+17)+2i+222t+22i+33(11t+11i+17)(11t+11i+18)),

Π1*(D[n])=(22t+8)2(22t+9)4(22+11)2(22t+13)2(22t+15)4(22t+17)4(22t+19)8(22t+21)4(22t+23)4(22t+25)4(22t+27)4(22t+29)4(22t+31)4(22t+33)4(22t+35)4Πi=1t1(22t+22i15)2i+2(22t+22i157)2i+2(22t+22i19)2i+2(22t+22i21)2i+2(22t+22i+23)2i+2(22t+22i+25)2i+2(22t+22i+27)2i+2(22t+22i29)2i+2(22t+22i+31)2i+2(22t+22i+33)2i+2(22t+22i+35)2i+2),

Π2*(D[n])=(11t+4)2(11t+56)6(11+6)4(11t+7)6(22t+8)8(22t+9)12(11t+10)12(11t+11)8(11t+12)8(11t+13)8(11t+14)8(11t+15)8(11t+16)8(11t+17)4(11t+18)4Πi=1t1(11t+11i7)2i+2(11t+11i8))2i+3(11t+11i19)32i+2(11t+11i10)32i+2(11t+11i11)2i+3(11t+11i+12)2i+3(11t+11i+13)2i+3(11t+11i+14)2i+3(11t+11i15)2i+3(11t+11i+16)2i+3(11t+11i+17)2i+3(11t+11i+18)2i+2),

Zg4(D[n])=2x22t+8+4x22t+9+2x22t+11+2x22t+13+4x22t+15+4x22t+17+8x22t+19+4x22t+21+4x22t+23+4x22t+25+4x22t+27+4x22t+29+4x22t+31+4x22t+33+4x22t+35+i=1t1(2i+2x22t+22i+15+2i+2x22t+22i+17+2i+3x22t+22i+19+2i+2x22t+22i+21+2i+2x22t+22i+23+2i+2x22t+22i+25+2i+2x22t+22i+27+2i+2x22t+22i+29+2i+2x22t+22i+31+2i+2x22t+22i+33+2i+2x22t+22i+35),

Zg6(D[n])=2x(11t+4)(11t+4)+4x(11t+4)(11t+5)+2x(11t+5)(11t+6)+2x(11t+6)(11t+7)+4x(11t+7)(11t+8)+4x(11t+8)(11t+9)+8x(11t+9)(11t+10)+4x(11t+10)(11t+11)+4x(11t+11)(11t+12)+4x(11t+12)(11t+13)+4x(11t+13)(11t+14)+4x(11t+14)(11t+15)+4x(11t+15)(11t+16)+4x(11t+16)(11t+17)+4x(11t+17)(11t+18)+i=1t1(2i+2x(11t+11i+7)(11t+11i+8)+2i+2x(11t+11i+8)(11t+11i+9)+2i+3x(11t+11i+9)(11t+11i+10)+2i+2x(11t+11i+10)(11t+11i+11)+2i+2x(11t+11i+11)(11t+11i+12)+2i+2x(11t+11i+12)(11t+11i+13)+2i+2x(11t+11i+13)(11t+11i+14)+2i+2x(11t+11i+14)(11t+11i+15)+2i+2x(11t+11i+15)(11t+11i+16)+2i+2x(11t+11i+16)(11t+11i+17)+2i+2x(11t+11i+17)(11t+11i+18)).

Proof. Since D[n] is symmetrical, we can mark the vertices several representative symbols which are described in Figure 3 and Figure 4. Next, we only present the detailed proof of GA4 index, and other parts of result can be yielded in the similar way.

If n ≡ 1(mod2), then let t=n12 and 1 ≤ it − 1. By the analysis of graph structure of D[n], the set of E(D[n]) can be divided into the following subsets which are described as follows:

  • (u, v): with eccentricities 11t + 4 and 11t + 4, and there are two edges in this class;
  • (v, w): with eccentricities 11t + 4 and 11t + 5, and there are four edges in this class;
  • (w, x): with eccentricities 11t + 5 and 11t + 6, and there are two edges in this class;
  • (x, y): with eccentricities 11t + 6 and 11t + 7, and there are two edges in this class;
  • (y, a1): with eccentricities 11t + 7 and 11t + 8, and there are four edges in this class;
  • (a1, b1): with eccentricities 11t + 8 and 11t + 9, and there are four edges in this class;
  • (b1, c1): with eccentricities 11t + 9 and 11t + 10, and there are eight edges in this class;
  • (c1, e1): with eccentricities 11t + 10 and 11t + 11, and there are four edges in this class;
  • (e1, f1): with eccentricities 11t + 11 and 11t + 12, and there are four edges in this class;
  • ( f1, g1): with eccentricities 11t + 12 and 11t + 13, and there are four edges in this class;
  • (g1, h1): with eccentricities 11t + 13 and 11t + 14, and there are four edges in this class;
  • (h1, v1): with eccentricities 11t + 14 and 11t + 15, and there are four edges in this class;
  • (v1, w1): with eccentricities 11t + 15 and 11t + 16, and there are four edges in this class;
  • (w1, x1): with eccentricities 11t + 16 and 11t + 17, and there are four edges in this class;
  • (x1, y1): with eccentricities 11t + 17 and 11t + 18, and there are four edges in this class;
  • (yi, ai+1): with eccentricities 11t + 11i + 7 and 11t + 11i + 8, and there are 2i+2 edges in this class;
  • (ai+1, bi+1): with eccentricities 11t + 11i + 8 and 11t + 11i + 9, and there are 2i+2 edges in this class;
  • (bi+1, ci+1): with eccentricities 11t + 11i + 9 and 11t + 11i + 10, and there are 2i+3 edges in this class;
  • (ci+1, ei+1): with eccentricities 11t + 11i + 10 and 11t + 11i + 11, and there are 2i+2 edges in this class;
  • (ei+1, fi+1): with eccentricities 11t + 11i + 11 and 11t + 11i + 12, and there are 2i+2 edges in this class;
  • ( fi+1, gi+1): with eccentricities 11t + 11i + 12 and 11t + 11i + 13, and there are 2i+2 edges in this class;
  • (gi+1, hi+1): with eccentricities 11t + 11i + 13 and 11t + 11i + 14, and there are 2i+2 edges in this class;
  • (hi+1, vi+1): with eccentricities 11t + 11i + 14 and 11t + 11i + 15, and there are 2i+2 edges in this class;
  • (vi+1, wi+1): with eccentricities 11t + 11i + 15 and 11t + 11i + 16, and there are 2i+2 edges in this class;
  • (wi+1, xi+1): with eccentricities 11t + 11i + 16 and 11t + 11i + 17, and there are 2i+2 edges in this class;
  • (xi+1, yi+1): with eccentricities 11t + 11i + 17 and 11t + 11i + 18, and there are 2i+2 edges in this class.

If t = 0, then n = 1, we have

GA4(D[1])=uvE(D[1])2ec(u)ec(v)ec(u)+ec(v)=22444+4+42454+5+22565+6+22676+7.

If t = 1, then n = 3, we get

GA4(D[3])=uvE(D[3])2ec(u)ec(v)ec(u)+ec(v)=22151515+15+42151615+16+22161716+17+22171817+18+42181918+19+42192019+20+42202120+21+42212221+22+42222322+23+42232423+24+42242524+25+42252625+26+42262726+27+42272827+28+42282928+29.

If n ≥ 5, then we obtain

GA4(D[n])=uvE(D[n])2ec(u)ec(v)ec(u)+ec(v)=22(11t+4)(11t+4)(11t+4)+(11t+4)+42(11t+4)(11t+5)(11t+4)+(11t+5)+22(11t+5)(11t+6)(11t+5)+(11t+6)+22(11t+6)(11t+7)(11t+6)+(11t+7)+42(11t+7)(11t+8)(11t+7)+(11t+8)+42(11t+8)(11t+9)(11t+8)+(11t+9)+82(11t+9)(11t+10)(11t+9)+(11t+10)+42(11t+10)(11t+11)(11t+10)+(11t+11)+42(11t+11)(11t+12)(11t+11)+(11t+12)+42(11t+12)(11t+13)(11t+12)+(11t+13)+42(11t+13)(11t+14)(11t+13)+(11t+14)+42(11t+14)(11t+15)(11t+14)+(11t+15)+42(11t+15)(11t+16)(11t+15)+(11t+16)+42(11t+16)(11t+17)(11t+16)+(11t+17)+42(11t+17)(11t+18)(11t+17)+(11t+18)+i=1t1(2i+22(11t+11i+7)(11t+11i+8)(11t+11i+7)+(11t+11i+8)+2i+22(11t+11i+8)(11t+11i+9)(11t+11i+8)+(11t+11i+9)+2i+32(11t+11i+9)(11t+11i+10)(11t+11i+9)+(11t+11i+10)+2i+22(11t+11i+10)(11t+11i+11)(11t+11i+10)+(11t+11i+11)+2i+22(11t+11i+11)(11t+11i+12)(11t+11i+11)+(11t+11i+12)+2i+22(11t+11i+12)(11t+11i+13)(11t+11i+12)+(11t+11i+13)+2i+22(11t+11i+13)(11t+11i+14)(11t+11i+13)+(11t+11i+14)+2i+22(11t+11i+14)(11t+11i+15)(11t+11i+14)+(11t+11i+15)+2i+22(11t+11i+15)(11t+11i+16)(11t+11i+15)+(11t+11i+16)+2i+22(11t+11i+16)(11t+11i+17)(11t+11i+16)+(11t+11i+17)+2i+22(11t+11i+17)(11t+11i+18)(11t+11i+17)+(11t+11i+18)).

If n ≡ 0(mod2), then let t=n2 and 1 ≤ it − 1. According to the analysis of molecular structure of D[n], the edge set of D[n] can be divided into the following subsets which are presented as follows:

  • (u, v): with eccentricities 11t and 11t, and there are two edges in this class;
  • (v, w): with eccentricities 11t and 11t + 1, and there are four edges in this class;
  • (w, x): with eccentricities 11t + 1 and 11t + 2, and there are two edges in this class;
  • (x, y): with eccentricities 11t + 2 and 11t + 3, and there are two edges in this class;
  • (y, a1): with eccentricities 11t + 3 and 11t + 4, and there are four edges in this class;
  • (ai, bi): with eccentricities 11t + 11i − 7 and 11t + 11i − 6, and there are 2i+1 edges in this class;
  • (bi, ci): with eccentricities 11t + 11i − 6 and 11t + 11i − 5, and there are 2i+2 edges in this class;
  • (ci, ei): with eccentricities 11t + 11i − 5 and 11t + 11i − 4, and there are 2i+1 edges in this class;
  • (ei, fi): with eccentricities 11t + 11i − 4 and 11t + 11i − 3, and there are 2i+1 edges in this class;
  • (fi, gi): with eccentricities 11t + 11i − 3 and 11t + 11i − 2, and there are 2i+1 edges in this class;
  • (gi, hi): with eccentricities 11t + 11i − 2 and 11t + 11i − 1, and there are 2i+1 edges in this class;
  • (hi, vi): with eccentricities 11t + 11i − 1 and 11t + 11i, and there are 2i+1 edges in this class;
  • (vi, wi): with eccentricities 11t + 11i and 11t + 11i + 1, and there are 2i+1 edges in this class;
  • (wi, xi): with eccentricities 11t + 11i + 1 and 11t + 11i + 2, and there are 2i+1 edges in this class;
  • (xi, yi): with eccentricities 11t + 11i + 2 and 11t + 11i + 3, and there are 2i+1 edges in this class;
  • (yi, ai+1): with eccentricities 11t + 11i + 3 and 11t + 11i + 4, and there are 2i+2 edges in this class;
  • (at, bt): with eccentricities 22t − 7 and 22t − 6, and there are 2t+1 edges in this class;
  • (bt, ct): with eccentricities 22t − 6 and 22t − 5, and there are 2t+2 edges in this class;
  • (ct, et): with eccentricities 22t − 5 and 22t − 4, and there are 2t+1 edges in this class;
  • (et, ft): with eccentricities 22t − 4 and 22t − 3, and there are 2t+1 edges in this class;
  • (ft, gt): with eccentricities 22t − 3 and 22t − 2, and there are 2t+1 edges in this class;
  • (gt, ht): with eccentricities 22t − 2 and 22t − 1, and there are 2t+1 edges in this class.

If t = 1, then n = 2, we have

GA4(D[2])=uvE(D[2])2ec(u)ec(v)ec(u)+ec(v)=22111111+11+42111211+12+22121312+13+22131413+14+42141514+15+42151615+16+82161716+17+42171817+18+42181918+19+42192019+20+42202120+21.

If n ≥ 4, then we obtain

GA4(D[n])=uvE(D[n])2ec(u)ec(v)ec(u)+ec(v)=22(11t)(11t)(11t)+(11t)+42(11t)(11t+1)(11t)+(11t+1)+22(11t+1)(11t+2)(11t+1)+(11t+2)+22(11t+2)(11t+3)(11t+2)+(11t+3)+42(11t+3)(11t+4)(11t+3)+(11t+4)+i=1t1(2i+12(11t+11i7)(11t+11i6)(11t+11i7)+(11t+11i6)+2i+22(11t+11i6)(11t+11i5)(11t+11i6)+(11t+11i5)+2i+12(11t+11i5)(11t+11i4)(11t+11i5)+(11t+11i4)+2i+12(11t+11i4)(11t+11i3)(11t+11i4)+(11t+11i3)+2i+12(11t+11i3)(11t+11i2)(11t+11i3)+(11t+11i2)+2i+12(11t+11i2)(11t+11i1)(11t+11i2)+(11t+11i1)+2i+12(11t+11i1)(11t+11i)(11t+11i1)+(11t+11i)+2i+12(11t+11i)(11t+11i+1)(11t+11i)+(11t+11i+1)+2i+12(11t+11i+1)(11t+11i+2)(11t+11i+1)+(11t+11i+2)+2i+12(11t+11i+2)(11t+11i+3)(11t+11i+2)+(11t+11i+3)+2i+12(11t+11i+3)(11t+11i+4)(11t+11i+3)+(11t+11i+4))+2t+12(22t7)(22t6)(22t7)+(22t6)+2t+22(22t6)(22t5)(22t6)+(22t5)+2t+12(22t5)(22t4)(22t5)+(22t4)+2t+12(22t4)(22t3)(22t4)+(22t3)+2t+12(22t3)(22t2)(22t3)+(22t2)+2t+12(22t2)(22t1)(22t2)+(22t1).

Thus, we yield the expected result.

3 Conclusion

In this paper, we mainly discuss two molecular graphs which commonly appeared in drug structures. Our results can be concluded as follows: first, we manifest the generalized degree distance of starlike tree; second, we report the eccentricity related indices of hetrofunctional dendrimer. Since these indices are widely used in the analysis of chemical and pharmacological properties of drugs, the theoretical results obtained in our article admit promising prospects of engineering applications in the field of chemistry, pharmacy and medical science.

4 Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Communicated by Wei Gao
Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper. The research is partially supported by NSFC (nos. 11401519, 11371328, and 11471293), the National Nature Science Fund Project (61262071), National Science and Technology Support Program(2013BAJ07B02), and Key Project of Applied Basic Research Program of Yunnan Province (2016FA024).

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  • [1]

    M. R. Farahani M. K. Jamil and M. Imran (2016) Vertex PIv Topological Index of Titania Carbon Nanotubes TiO2(mn) Applied Mathematics and Nonlinear Sciences 1 No 1 175-182.

    • Crossref
    • Export Citation
  • [2]

    M. K. Jamil M. R. Farahani M. Imran and M. A. Malik (2016) Computing Eccentric Version of Second Zagreb Index of Polycyclic Aromatic Hydrocarbons PAHk Applied Mathematics and Nonlinear Sciences 1 No 1 247-252.

    • Crossref
    • Export Citation
  • [3]

    W. Gao W. Wang M. K. Jamil and M. R. Farahani (2016) Electron Energy Studying of Molecular Structures via Forgotten Topological Index Computation Journal of Chemistry Volume 2016 Article ID 1053183 7 pages.

    • Crossref
    • Export Citation
  • [4]

    W. Gao M. R. Farahani and M. K. Jamil (2016) The eccentricity version of atom-bond connectivity index of linear polycene parallelogram benzenoid ABC5(P(nn)) Acta Chimica Slovenica 63 No 2 376-379.

    • Crossref
    • Export Citation
  • [5]

    W. Gao W. Wang and M. R. Farahani (2016) Topological Indices Study of Molecular Structure in Anticancer Drugs Journal of Chemistry Volume 2016 Article ID 3216327 8 pages.

    • Crossref
    • Export Citation
  • [6]

    W. Gao M. R. Farahani and L. Shi (2016) The forgotten topological index of some drug structures Acta Medica Mediterranea 32 579-585.

  • [7]

    W. Gao M. K. Siddiqui M. Imran M. K. Jamil and M. R. Farahani (2016) Forgotten topological index of chemical structure in drugs Saudi Pharmaceutical Journal 24 No 3 258-264.

    • Crossref
    • Export Citation
  • [8]

    W. Gao and W. Wang (2014) Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures Journal of Chemistry Volume 2014 Article ID 906254 8 pages.

    • Crossref
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