ISRN.APPLIED.MATHEMATICS ISRN Applied Mathematics 2090-5572 Hindawi Publishing Corporation 781648 10.1155/2014/781648 781648 Research Article Some Properties on the Harmonic Index of Molecular Trees Liu Shaoqiang http://orcid.org/0000-0001-7034-9062 Li Jianxi Kearsley A. J. Liu X. 1 School of Mathematics and Statistics Minnan Normal University Zhangzhou, Fujian 363000 China mnnu.net 2014 2912014 2014 04 10 2013 04 12 2013 29 1 2014 2014 Copyright © 2014 Shaoqiang Liu and Jianxi Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The harmonic index of a graph G is defined as the sum of weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u in G. In this paper, some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are provided, respectively.

1. Introduction

Let G be a simple graph with vertex set V(G) and edge set E(G). Its order is |V(G)|, denoted by n. Let d(v) and N(v) be the degree and the set of neighbors of vV(G), respectively. The harmonic index of G is defined in  as (1)H(G)=uvE(G)2d(u)+d(v), where the summation goes over all edges uv of G. This index was extensively studied recently. For example, Zhong [2, 3] and Zhong and Xu  determined the minimum and maximum values of the harmonic index for simple connected graphs, trees, unicyclic graphs, and bicyclic graphs, respectively. Some upper and lower bounds on the harmonic index of a graph were obtained by Ilic . Xu  and Deng et al. [7, 8] established some relationship between the harmonic index of a graph and its topological indices, such as Randić index, atom-bond connectivity index, chromatic number, and radius, respectively. Wu et al.  determined the graph with minimum harmonic index among all the graphs (or all triangle-free graphs) with minimum degree at least two. More information on the harmonic index of a graph can be found in .

The general sum-connectivity index of G was proposed by Du et al. in  and defined as (2)χα(G)=uvE(G)(d(u)+d(v))α. Clearly, H(G)=2χ-1(G). Du et al.  determined the maximum value and the corresponding extremal trees for the general sum-connectivity indices of trees for α<α0, where α0=-4.3586 is the unique root of the equation (4α-5α)/(5α-6α)=3. However, they did not consider the general sum-connectivity indices with α=-1.

A molecular tree T is a tree with maximum degree at most four. It models the skeleton of an acyclic molecule . As far as we know, the mathematical properties of related indices for molecular trees have been studied extensively. For example, Gutman et al. [13, 14] determined the molecular trees with the first maximum, the second maximum, and the third maximum Randić indices, respectively. Du et al.  further determined the fourth maximum Randić index for molecular trees. Li et al. [16, 17] obtained the lower and upper bounds for the general Randić index R-1 for molecular trees and determined the molecular tree with minimum general Randić index among molecular trees with given pendant vertices. The graphs with maximum and minimum sum-connectivity indices among molecular trees with given pendant vertices were determined in Xing et al. .

In this paper, we consider the similar problem of determining the graphs with maximum or minimum harmonic index for molecular trees. Some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are determined, respectively.

2. Properties of the Harmonic Index for Molecular Trees

In this section, some general properties of the harmonic index for molecular trees are explored. Before this, some notations are needed. Let 𝒯(n) be the set of molecular trees of order n. For T𝒯(n), denote by ni the number of vertices with degree i for i=1,2,3,4, and denote by xij the number of edges in T that connect vertices of degree i and j, where 1ij4. Obviously, n1 is the number of pendant vertices. Note that x11=0. Then we have (3)n1+n2+n3+n4=n,n1+2n2+3n3+4n4=2(n-1),x12+x13+x14=n1,x12+2x22+x23+x24=2n2,x13+x23+2x33+x34=3n3,x14+x24+x34+2x44=4n4,(4)H(T)=23x12+12x13+25x14+12x22+25x23+13x24+13x33+27x34+14x44.

Moreover, Gutman and Miljković in  established the following relations: (5)x14=2n+23-43x12-109x13-23x22-49x23-13x24-29x33-19x34,x44=n-53+13x12+19x13-13x22-59x23-23x24-79x33-89x34. Substituting these equations into (4), we have (6)H(T)=7n20-320+1360x12+112x13+320x22+112x23+130x24+120x33+2105x34.

Now, let (7)φ(T)=H(T)-(7n20-320)=1360x12+112x13+320x22+112x23+130x24+120x33+2105x34. Note that from (3), we have n2=(1/2)x12+x22+(1/2)x23+(1/2)x24 and n3=(1/3)x13 + (1/3)x23+(2/3)x33+(1/3)x34. Then we have the following lemma.

Lemma 1.

For any T𝒯(n), if n2+n33, then φ(T)72/420.

Proof.

We consider the following two cases.

Case 1  (n2+n3>3). Note that (8)φ(T)=1360x12+112x13+320x22+112x23+130x24+120x33+2105x34=1360x12+340×(2x22)+124x23+130x24+112x13+124x23+140×(2x33)+2105x342n2×130+3n3×210596420sincen2+n3>3.

Thus φ(T)96/420>72/420.

Case 2 (n2+n3=3). If n2=3 and n3=0, then (9)φ(T)=1360x12+320x22+130x24=1360x12+340×(2x22)+130x246×130=84420>72420sincex12+2x22+x24=6.

If n2=2 and n3=1, then (10)φ(T)=1360x12+112x13+320x22+112x23+130x24+120x33+2105x34=1360x12+340×(2x22)+124x23+130x24+112x13+124x23+140×(2x33)+2105x344×130+3×2105=80420>72420, since x12+2x22+x23+x24=4 and x13+x23+2x33+x34=3.

If n2=1 and n3=2, then (11)φ(T)=1360x12+112x13+320x22+112x23+130x24+120x33+2105x34=1360x12+340×(2x22)+124x23+130x24+112x13+124x23+140×(2x33)+2105x342×130+6×2105=76420>72420, since x12+2x22+x23+x24=2 and x13+x23+2x33+x34=6.

If n2=0 and n3=3, then (12)φ(T)=112x13+120x33+2105x34=112x13+140×(2x33)+2105x349×2105=72420,sincex13+2x33+x34=9.   This completes the proof.

Lemma 2.

For any T𝒯(n), if n2+n32, then φ(T)<72/420.

Proof.

If n2+n32, then the graphically feasible combinations of x12,x13,x22, x23,x24,x33, and x34 for which φ(T)<72/420 are listed in Table 1, where nk(mod3) and corresponding nine classes of molecular trees are denoted by i𝒯(n) for i=1,2,,9, respectively.

Nine classes of molecular trees and φ(T)<72/420.

Set n 2 n 3 Nonzero xijs value φ ( T ) k n
1 T ( n ) 0 0 0 2 n 5
2 T ( n ) 0 1 x 34 = 3 24 / 420 1 n 13
3 T ( n ) 1 0 x 24 = 2 28 / 420 0 n 9
4 T ( n ) 0 2 x 34 = 6 48 / 420 0 n 21
5 T ( n ) 0 1 x 34 = 2 , x 13 = 1 51 / 420 1 n 10
6 T ( n ) 0 2 x 34 = 4 , x 33 = 1 53 / 420 0 n 18
7 T ( n ) 1 1 x 24 = 2 , x 34 = 3 52 / 420 2 n 17
8 T ( n ) 2 0 x 24 = 4 56 / 420 1 n 13
9 T ( n ) 1 1 x 24 = 1 , x 34 = 2 , x 23 = 1 65 / 420 2 n 14

Note that the smaller φ(T), the smaller H(T). Then by Lemmas 1 and 2, we have the following properties for 𝒯(n).

Theorem 3.

For n0(mod3), if T𝒯(n), then

when n9, H(T)(7n/20)-(1/12), the equality holds if and only if T3T(n);

when n21 and T3T(n), H(T)(7n/20)-(1/28), the equality holds if and only if T4T(n);

when n21 and T3T(n)4T(n), H(T)(7n/20)-(1/42), the equality holds if and only if T6T(n).

Theorem 4.

For n1(mod3), if T𝒯(n), then

when n13, H(T)(7n/20)-(13/140), the equality holds if and only if T2T(n);

when n13 and T2T(n), H(T)(7n/20)-(1/35), the equality holds if and only if T5T(n)(when n=10, H(T)=(7n/20)-(1/35), and T5T(n));

when n13 and T2T(n)5T(n), H(T)(7n/20)-(1/60), the equality holds if and only if T8T(n).

Theorem 5.

For n2(mod3), if T𝒯(n), then

when n5, H(T)(7n/20)-(3/20), the equality holds if and only if T1T(n);

when n17 and T1T(n), H(T)(7n/20)-(11/420), the equality holds if and only if T7T(n);

when n17 and T1T(n)7T(n), H(T)(7n/20)+(1/210), the equality holds if and only if T9T(n).

In , Ilic deduced that by removing an edge with the minimal weight from a graph, where the weight of e=uv is denoted by 2/(d(v)+d(u)), its harmonic index strictly decreases. For molecular tree T, by removing any pendent vertex from T, we have the following theorem:

Theorem 6.

Let T be a molecular tree of order n3, and let v be a pendent vertex of  T. Then one has H(T-v)<H(T).

Proof.

Let e=uvE(T) be a pendent edge, where d(v)=1 and d(u)2. Now we consider the difference Δ=H(T)-H(T-v) in the following three cases.

Case 1 (d(u)=2 and N(u)={v1,v}). The result follows from (13)Δ=2d(v1)+2+23-2d(v1)+1=23-2(2+d(v1))(d(v1)+1)23-2(2+1)(1+1)=13.

Case 2 (d(u)=3 and N(u)={v1,v2,v}). The result follows from (14)Δ=2d(v1)+3+2d(v2)+3+24-2d(v1)+2-2d(v2)+2=12-2(3+d(v1))(d(v1)+2)-2(3+d(v2))(d(v2)+2)12-22(3+1)(2+1)=16.

Case 3 (d(u)=4 and N(u)={v1,v2,v3,v}). The result follows from (15)Δ=2d(v1)+4+2d(v2)+4+2d(v3)+4+25-2d(v1)+3-2d(v2)+3-2d(v3)+3=25-2(3+d(v1))(d(v1)+4)-2(3+d(v2))(d(v2)+4)-2(3+d(v3))(d(v3)+4)25-32(3+1)(4+1)=110. The proof is completed.

Remark 7.

Note that removing a pendent vertex is equal to removing an pendent edge. Thus Theorem 6 states that, for molecular trees, the removed pendent vertex may not be located at an edge with the minimal weight, which is illustrated by the following example. In Figure 1, the weight of e1 and that of e2 are 2/(d(u)+d(v1))=2/(1+3)=1/2 and 2/(d(v2)+d(v3))=2/(3+3)=1/3, respectively. But H(T-e1)=H(T-u)<H(T).

An example of T-u.

T, H(T) = 56/15

Tu, H(Tu) = 52/15

3. Smallest Values of Harmonic Index for Molecular Trees with Given Pendent Vertices

In this section, the smallest values of harmonic index for molecular trees with given number of pendent vertices are determined.

Theorem 8.

Let T be a tree of order n with p pendant vertices, where 2pn-2. Then (16)H(T)2p+2-4p+1+n-p2+53, the equality holds if and only if TSn,p, where Sn,p (shown in Figure 2) is a tree obtained by attaching p-1 pendent vertices to an end vertex of the path Pn-p+1.

The tree Sn,p.

Proof.

If p=2, then TPn; the result is obvious. For 3pn-2, we prove the theorem by induction on n. If n=5, then TS5,3 or TS5; the result is obvious. Suppose that n6. Let u be a pendant vertex and N(u)={v}. Now we consider the d(v) in the following two cases.

Case 1  (d(v)=2). Then T-u contains p pendant vertices. For n6, there always exists ωN(v){u} such that d(w)2. Thus (17)H(T)-H(T-u)=2d(w)+2+23-2d(w)+124+23-23=12. The equality holds if and only if d(w)=2.

If p=n-2, then T=Sn,n-2; that is, T-u=Sn-1,n-2. Hence d(w)>2; if 3pn-3 and T-u=Sn-1,p, then d(w)=2. By the induction hypothesis, we have (18)H(T)H(T-u)+122p+2+2(p-1)p+1+23+2(n-p-3)4+12=2p+2-4p+1+n-p2+53. The equality holds if and only if T-u=Sn-1,p; that is, TSn,p.

Case 2 (d(v)3). Then T-u contains p-1 pendant vertices, and N(v){u} contains some vertices with degree at least two. For 3pn-2, we have (19)H(T)-H(T-u)=2d(v)+1-wN(v)u(2d(w)+d(v)-1-2d(w)+d(v))2d(v)+1-(2d(v)-1+2-2d(v)+2)-(d(v)-2)(2d(v)-1+1-2d(v)+1)=2(1d(v)+2-3d(v)+1+2d(v)). The equality holds if and only if N(v) contains one vertex of degree two and d(v)-1 vertices of degree one. Let g(x)=2((1/(x+2))-(3/(x+1))+(2/x)). Note that g(x)=-(((4x3+30x2+48x+16))(((x+2)2(x+1)2x2)))<0 for x0. Then g(x) is strictly decreasing on x0. Recall that d(v)p. Hence we have (20)H(T)-H(T-u)2(1p+2-3p+1+2p). The equality holds if and only if N(v) contains one vertex of degree two and d(v)-1 vertices of degree one; that is, T=Sn,p and d(v)=p. By the induction hypothesis, we have (21)H(T)H(T-u)+2(1p+2-3p+1+2p)2p+1-4p+n-p2+53+2(1p+2-3p+1+2p)=2p+2-4p+1+n-p2+53. The equality holds if and only if T-uSn-1,p-1 and d(v)=p; that is, TSn,p.

Lemma 9.

Let n,p be positive integers with 2pn-2. Let (22)f(n,p)=2p+2-4p+1+n-p2+53. Then f(n,p) is monotonically decreasing on p.

Proof.

We consider the derivative of f(n,p). For p2, we have (23)f(n,p)p=-2(p+2)2+4(p+1)2-12=-p4-6p3-9p2+12p+242(p+1)2(p+2)2.

Let g(p)=-p4-6p3-9p2+12p+24. Clearly, g(2)<0. We consider the derivative of g(p). For p2, we have (24)g(p)=-4p3-18p2-18p+12<0.

Thus g(p) is monotonically decreasing on p and g(p)g(2)<0. That is, f(n,p)/p<0 for 2pn-2. Hence f(n,p) is monotonically decreasing on p.

Recall that if p=n-1, then Sn,n-1=Sn and H(Sn)=2(n-1)/n. Moreover, by Lemma 9, we have f(n,p)f(n,n-2)=(2/n)-(4/(n-1))+1+(5/3)>2(n-1)/n=H(Sn). This together with Theorem 8 implies the following

Corollary 10 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

Among all trees of order n, the minimum harmonic index is attained uniquely by the star Sn.

Let 𝒯n,p be the set of molecular trees of order n with p pendent vertices. Now we introduce two classes of molecular trees of order n with p pendent vertices.

The first class is denoted by e(n,p) for even p with 6p(n+3)/2 (shown in Figure 3). Those trees are composed of (p-2)/2 star S5, which are connected by paths whose lengths may be zero. Note that n1=p,n2=n-(3p/2)+1,n3=0,n4=(p/2)-1,x14=p,x22=n-2p+3, and x24=p-4.

An example of molecular tree in e(n,p).

The second class is denoted by l(n,p) for odd p with 9p(n+2)/2 (shown in Figure 4). Those trees are composed of (p-3)/2 star S5 and one star S4, which are connected by paths whose lengths may be zero, and the unique star S4 is connected by three stars S5. Note that n1=p, n2=n-((3p+1)/2)+1, n3=1, n4=((p-1)/2)-1, x14=p, x24=p-6, x22=n-2p+2, and x23=3.

An example of molecular tree in l(n,p).

For p4, if 3n5, then Corollary 10 implies that Sn is the unique molecular tree with the minimum harmonic index; if n6, then Theorem 8 and Lemma 9 imply that Sn,4 is the unique molecular tree with the minimum harmonic index. If p5, then Sn,p is not a molecular tree. The following gives the smallest value for 𝒯n,p with p5.

Theorem 11.

Let T𝒯n,p and p5. Then (25)H(T)n2-415p+16; the equality holds if and only if Te(n,p) for even p with 6p(n+3)/2. Moreover, if p is odd and 9p(n+2)/2, then (26)H(T)n2-415p+15; the equality holds if and only if Tl(n,p).

Proof.

For T𝒯n,p, Xing et al.  deduced the following relations: (27)x14=p-x12-x13,x22=n-2p-x12-13x13-13x23+13x33+23x34+x44+3,x24=p+x12+13x13-23x23-43x33-53x34-2x44-4. Substituting these equations into (4), we have (28)H(T)=n2-415p+16+110x12+245x13+190x23+118x33+463x34+112x44. Clearly, the minimum value of H(T) is attained at x12=x13=x23=x33=x34=x44=0. That is H(T)(n/2)-(4/15)p+(1/6). Moreover if H(T)=(n/2)-(4/15)p+(1/6), then x14=p, x22=n-2p+3, x24=p-4, n2=n-(3p/2)+1, n3=0, and n4=(p/2)-1, implying that Te(n,p).

Now suppose that not all of x12, x13, x23, x33, x34, and x44 in (28) are zero. Let (29)φ(T)=H(T)-(n2-415p+16)=110x12+245x13+190x23+118x33+463x34+112x44.

Now we consider the following two cases.

Case 1 (x12+x13+x33+x34+x44>0). Then φ(T)min{(1/10),(2/45),(1/18),(4/63),(1/12)} = (2/45)>(1/30), the result holds.

Case 2  (x12=x13=x33=x34=x44=0 and x230). Clearly, x23=3n3 since x13+x23+2x33+x34=3n3. Thus the only possible combination of x12, x13, x23, x33, x34, and x44 for which φ(T)(1/30) is x12=x13=x33=x34=x44=0, x23=3 with n3=1.

Hence the minimum value of H(T) is attained at x12=x13=x33=x34=x44=0,x23=3. That is, H(T)(n/2)-(4/15)p+(1/5). Moreover if H(T)=(n/2)-(4/15)p+(1/5), then x14=p, x22=n-2p+2, x24=p-6, n2=n-((3p+1)/2)+1, n3=1, and n4=((p-1)/2)-1, implying that Tl(n,p). This completes the proof.

4. Largest Values for Harmonic Index of Molecular Trees with Given Pendent Vertices

In this section, largest values for the harmonic indices of molecular trees with given number of pendent vertices are determined.

For T𝒯n,p, let Vi(T)={v:vV(T),d(v)=i} and Ei,j(T)={e:e=uvE(T),d(u)=i,d(v)=j}.

Let 3(n,p) (shown in Figure 5) be a class of molecular trees of order n. For this type of molecular trees, there are p-2 vertices of maximal degree three, which induce a tree and any of these vertices is adjacent to either another vertex of degree three or a vertex of degree two. Note that n1=p, n2=n-2p+2, n3=p-2, n4=0, x12=x23=p, x13=0, x22=n-3p+2, and x33=p-3. Clearly, 3p(n+2)/3.

An example of L3(n,p).

Lemma 12.

Let T be a tree with maximum harmonic index among all trees in 𝒯n,p. Then either E22(T)= or V4(T)E13(T)=.

Proof.

We shall prove the contrapositive of the lemma. If E22(T) and V4(T)E13(T), then either E22(T) and V4(T), or E22(T) and E13(T).

If E22(T) and V4(T), then there exists a vertex vV4(T) and N(v) has at most one neighbor of degree four. Let T be the tree obtained from T by contracting the edge eE22(T) and splitting the vertex v into (v1,v2). Clearly, T𝒯n,p and (30)H(T)-H(T)=23+3-22+2+wN(v)(23+d(w)-24+d(w))=-16+wN(v)(23+d(w)-24+d(w))>-16+3(23+3-24+3)+23+4-24+4=184>0; if E22(T) and E13(T), suppose that uvE13(T), d(u)=1 and d(v)=3. Let T be the tree obtained from T by contracting the edge eE22(T) and attaching an edge to u. Clearly, T𝒯n,p and (31)H(T)-H(T)=22+1+22+3-22+2-21+3=115>0. For each case, we have H(T)>H(T), which contradicts T with maximum harmonic index among all trees in 𝒯n,p. This completes the proof.

Theorem 13.

Let T𝒯n,p with p2. Then (32)H(T)n2-112p; the equality holds if and only if TPn.

Proof.

For any molecular tree T, the following relations were deduced by Xing et al. in : (33)x12=p-x13-x14,x22=n-52p+12x13+34x14-12x23-14x24+14x34+12x44+2,x33=32p-12x13-34x14-12x23-34x24-54x34-32x44-3. Substituting these equations into (4), we have (34)H(T)=n2-112p-112x13-17120x14-160x23-124x24-1168x34. Clearly, the maximum value of H(T) is attained at x13=x14=x23=x24=x34=0. That is, H(T)(n/2)-(1/12)p. Moreover, if H(T)=(n/2)-(1/12)p, then x12=p, x22=n-(5/2)p+2, x33=(3/2)p-3. Note that x33=x44=0 since T𝒯n,p; that is, p=2,x12=2, and x22=n-3. This implies that TPn.

Theorem 14.

Let T𝒯n,p(TPn) with |E23(T)|=p and |E22(T)|. Then (35)H(T)n2-110p; the equality holds if and only if T3(n,p).

Proof.

From the proof of Theorem 13, we can see that if TPn, then not all of x13, x14, x23, x24, and x34 in (34) are zero. By Lemma 12, if T with maximum harmonic index among all trees in 𝒯n,p and E22(T), then V4(T)E13(T)=. That is, x14=x24=x34=x44=x13=0. Theorem 13 implies that x12=p, x22=n-(5/2)p-(1/2)x23+2, and x33=(3/2)p-(1/2)x23-3. If |E23(T)|=p, then x23=p, x12=p, x22=n-3p+2, and x33=p-3. Note that p3 and |E23(T)|=p since TPn. Then H(T)(2p/3)+(2p/(2+3))+((2(p-3))/(3+3))+((2(n-3p+2))/(2+2))=(n/2)-(1/10)p; the equality holds if and only if T3(n,p). This completes the proof.

Conflict of Interests

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

Acknowledgments

This work is partially supported by NSF of China (no. 11101358), NSF of Fujian (nos. 2011J05014 and 2011J01026), and Project of Fujian Education Department (no. JA11165).

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