Abstract
The harmonic index of a graph is defined as the sum of weights of all edges of , where denotes the degree of the vertex in . 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 be a simple graph with vertex set and edge set . Its order is , denoted by . Let and be the degree and the set of neighbors of , respectively. The harmonic index of is defined in [1] as where the summation goes over all edges of . This index was extensively studied recently. For example, Zhong [2, 3] and Zhong and Xu [4] 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 [5]. Xu [6] 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. [9] 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 [10].
The general sum-connectivity index of was proposed by Du et al. in [11] and defined as Clearly, . Du et al. [11] determined the maximum value and the corresponding extremal trees for the general sum-connectivity indices of trees for , where is the unique root of the equation . However, they did not consider the general sum-connectivity indices with .
A molecular tree is a tree with maximum degree at most four. It models the skeleton of an acyclic molecule [12]. 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. [15] 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 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. [18].
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 be the set of molecular trees of order . For , denote by the number of vertices with degree for , and denote by the number of edges in that connect vertices of degree and , where . Obviously, is the number of pendant vertices. Note that . Then we have
Moreover, Gutman and Miljković in [14] established the following relations: Substituting these equations into (4), we have
Now, let Note that from (3), we have and + . Then we have the following lemma.
Lemma 1. For any , if , then .
Proof. We consider the following two cases.
Case 1 (). Note that
Thus .
Case 2(). If and , then
If and , then
since and .
If and , then
since and .
If and , then
This completes the proof.
Lemma 2. For any , if , then .
Proof. If , then the graphically feasible combinations of , , and for which are listed in Table 1, where and corresponding nine classes of molecular trees are denoted by for , respectively.
Note that the smaller , the smaller . Then by Lemmas 1 and 2, we have the following properties for .
Theorem 3. For , if , then(a)when , , the equality holds if and only if ;(b)when and , , the equality holds if and only if ;(c)when and , , the equality holds if and only if .
Theorem 4. For , if , then(a)when , , the equality holds if and only if ;(b)when and , , the equality holds if and only if when , , and ;(c)when and , , the equality holds if and only if .
Theorem 5. For , if , then(a)when , , the equality holds if and only if ;(b)when and , , the equality holds if and only if ;(c)when and , , the equality holds if and only if .
In [5], Ilic deduced that by removing an edge with the minimal weight from a graph, where the weight of is denoted by , its harmonic index strictly decreases. For molecular tree , by removing any pendent vertex from , we have the following theorem:
Theorem 6. Let be a molecular tree of order , and let be a pendent vertex of . Then one has .
Proof. Let be a pendent edge, where and . Now we consider the difference in the following three cases.
Case 1( and ). The result follows from
Case 2( and ). The result follows from
Case 3( and ). The result follows from
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 and that of are and , respectively. But .
(a) T, H(T) = 56/15
(b) T – u, H(T − u) = 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 be a tree of order with pendant vertices, where . Then the equality holds if and only if , where (shown in Figure 2) is a tree obtained by attaching pendent vertices to an end vertex of the path .
Proof. If , then ; the result is obvious. For , we prove the theorem by induction on . If , then or ; the result is obvious. Suppose that . Let be a pendant vertex and . Now we consider the in the following two cases.
Case 1 (). Then contains pendant vertices. For , there always exists such that . Thus
The equality holds if and only if .
If , then ; that is, . Hence ; if and , then . By the induction hypothesis, we have
The equality holds if and only if ; that is, .
Case 2(). Then contains pendant vertices, and contains some vertices with degree at least two. For , we have
The equality holds if and only if contains one vertex of degree two and vertices of degree one. Let . Note that for . Then is strictly decreasing on . Recall that . Hence we have
The equality holds if and only if contains one vertex of degree two and vertices of degree one; that is, and . By the induction hypothesis, we have
The equality holds if and only if and ; that is, .
Lemma 9. Let be positive integers with . Let Then is monotonically decreasing on .
Proof. We consider the derivative of . For , we have
Let . Clearly, . We consider the derivative of . For , we have
Thus is monotonically decreasing on and . That is, for . Hence is monotonically decreasing on .
Recall that if , then and . Moreover, by Lemma 9, we have . This together with Theorem 8 implies the following
Corollary 10 (see [5]). Among all trees of order , the minimum harmonic index is attained uniquely by the star .
Let be the set of molecular trees of order with pendent vertices. Now we introduce two classes of molecular trees of order with pendent vertices.
The first class is denoted by for even with (shown in Figure 3). Those trees are composed of star , which are connected by paths whose lengths may be zero. Note that , and .
The second class is denoted by for odd with (shown in Figure 4). Those trees are composed of star and one star , which are connected by paths whose lengths may be zero, and the unique star is connected by three stars . Note that , , , , , , , and .
For , if , then Corollary 10 implies that is the unique molecular tree with the minimum harmonic index; if , then Theorem 8 and Lemma 9 imply that is the unique molecular tree with the minimum harmonic index. If , then is not a molecular tree. The following gives the smallest value for with .
Theorem 11. Let and . Then the equality holds if and only if for even with . Moreover, if is odd and , then the equality holds if and only if .
Proof. For , Xing et al. [18] deduced the following relations:
Substituting these equations into (4), we have
Clearly, the minimum value of is attained at . That is . Moreover if , then , , , , , and , implying that .
Now suppose that not all of , , , , , and in (28) are zero. Let
Now we consider the following two cases.
Case 1 (). Then = , the result holds.
Case 2 ( and ). Clearly, since . Thus the only possible combination of , , , , , and for which is , with .
Hence the minimum value of is attained at . That is, . Moreover if , then , , , , , and , implying that . 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 , let and .
Let (shown in Figure 5) be a class of molecular trees of order . For this type of molecular trees, there are 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 , , , , , , , and . Clearly, .
Lemma 12. Let be a tree with maximum harmonic index among all trees in . Then either or .
Proof. We shall prove the contrapositive of the lemma. If and , then either and , or and .
If and , then there exists a vertex and has at most one neighbor of degree four. Let be the tree obtained from by contracting the edge and splitting the vertex into (). Clearly, and
if and , suppose that , and . Let be the tree obtained from by contracting the edge and attaching an edge to . Clearly, and
For each case, we have , which contradicts with maximum harmonic index among all trees in . This completes the proof.
Theorem 13. Let with . Then the equality holds if and only if .
Proof. For any molecular tree , the following relations were deduced by Xing et al. in [18]: Substituting these equations into (4), we have Clearly, the maximum value of is attained at . That is, . Moreover, if , then , , . Note that since ; that is, , and . This implies that .
Theorem 14. Let () with and . Then the equality holds if and only if .
Proof. From the proof of Theorem 13, we can see that if , then not all of , , , , and in (34) are zero. By Lemma 12, if with maximum harmonic index among all trees in and , then . That is, . Theorem 13 implies that , , and . If , then , , , and . Note that and since . Then ; the equality holds if and only if . 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).