1 Introduction
Abstract

The eccentric connectivity index is a distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum among trees with a fixed maximum vertex degree, and characterize such trees with minimum  . In addition, we propose a simple linear algorithm for calculating of trees.

.

Eccentric Connectivity Index

[12pt] of Chemical Trees




Aleksandar Ilić 333Corresponding author.

Faculty of Sciences and Mathematics, Višegradska 33, 18 000 Niš, Serbia

e-mail: aleksandari@gmail.com

Ivan Gutman

Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia

e-mail: gutman@kg.ac.rs

(Received May 25, 2009)

1 Introduction

Let be a simple connected graph with vertices. For a vertex  , denotes the degree of  . For vertices  , the distance is defined as the length of a shortest path between and in  . The eccentricity of a vertex is the maximum distance from to any other vertex.

Sharma, Goswami and Madan [14] introduced a distance–based molecular structure descriptor, which they named “eccentric connectivity index” and which they defined as

The index was successfully used for mathematical modeling of biological activities of diverse nature [2, 5, 10, 12, 13]. Some mathematical properties of were recently reported in [22].

Chemical trees (trees with maximum vertex degree at most four) provide the graph representation of alkanes [7]. It is therefore a natural problem to study trees with bounded maximum degree.

Denote by the maximum vertex degree of a tree  . The path is the unique -vertex tree with  , while the star is the unique tree with  . Therefore, we can assume that  .

For an arbitrary tree on vertices [22],

2 Chemical trees with maximum eccentric connectivity index

The broom is a tree consisting of a star and a path of length attached to a pendent vertex of the star. It is proven in [11] that among trees with maximum vertex degree equal to  , the broom uniquely minimizes the largest eigenvalue of the adjacency matrix. Further, within the same class of trees, the broom has minimum Wiener index and Laplacian-energy like invariant [16]. In [18] and [21] it was demonstrated that the broom has minimum energy among trees with, respectively, fixed diameter and fixed number of pendent vertices.

Figure 1: The broom  .

The -starlike tree is a tree composed of the root  , and the paths  ,  , …,  , attached to  . The number of vertices of is thus equal to  . Notice that the broom is a -starlike tree,  .

Theorem 2.1

Let be a vertex of a nontrivial connected graph  . For nonnegative integers and  , let denote the graph obtained from by attaching to the vertex pendent paths and of lengths and  , respectively. If  , then

Proof: The degrees of vertices and are changed, while all other vertices have the same degree in as in  . Since after this transformation the longer path has increased, the eccentricity of vertices from are either the same or increased by one. We will consider three cases based on the longest path from the vertex in the graph  . Denote by and the degree and eccentricity of vertex in  .

Case 1. The length of the longest path from the vertex in is greater than  . This means that the pendent vertex of  , most distant from is the most distant vertex for all vertices of and  . It follows that for all vertices  , while the eccentricity of increased by  .

Case 2. The length of the longest path from the vertex in is less than or equal to and greater than  . This means that either the vertex of that is most distant from or the vertex is the most distant vertex for all vertices of  , while for vertices the most distant vertex is  . It follows that for vertices  , while for vertices  . The eccentricity of increased by at least  .

Case 3. The length of the longest path from the vertex in is less than or equal to  . This means that the pendent vertex most distant from the vertices of and is either or  , depending on the position. Using the formula for eccentric connectivity index of a path, we have

Since is a nontrivial graph with at least one vertex, we have strict inequality.

This completes the proof.


Theorem 2.2

Let be an arbitrary tree on vertices with maximum vertex degree  . Then

Proof: Fix a vertex of degree as a root and let be the trees attached at  . We can repeatedly apply the transformation described in Theorem 2.1 to any vertex of degree at least three with greatest eccentricity from the root in every tree  , as long as does not become a path. When all trees turn into paths, we can again apply transformation from Theorem 2.1 at the vertex  as long as there exists at least two paths of length greater than one, further decreasing the eccentric connectivity index. Finally, we arrive at the broom as the unique tree with maximum eccentric connectivity index.


By direct verification, it holds

From the above proof, we also get that has the second minimal among trees with maximum vertex degree  .

It was proven in [22] that the path has maximum and the star minimum -value among connected graphs on vertices. From Theorem 2.2 we know that the maximum eccentric connectivity index among trees on  vertices is achieved for one of the brooms  . If  , we can apply the transformation from Theorem 2.1 at the vertex of degree  in and obtain  . Thus, it follows

Also, it follows that has the second maximum eccentric connectivity index among trees on vertices.

3 The minimum eccentric connectivity index of trees with fixed
radius

Vertices of minimum eccentricity form the center. A tree has exactly one or two adjacent center vertices; in this latter case one speaks of a bicenter. In what follows, if a tree has a bicenter, then our considerations apply to any of its center vertices.

For a tree with radius  ,

Let be the set of -vertex trees obtained from the path by attaching pendent vertices to and/or  , where  . Zhou and Du in [22] proved that for arbitrary tree on vertices and diameter  ,

with equality if and only if  . Using the transformation from Theorem 2.1 and applying it to a center vertex, it follows that for and  .

Corollary 3.1

Let be an arbitrary tree on vertices with radius  . Then

with equality if and only if  .

4 The maximum eccentric connectivity index of trees with
perfect matchings

A graph possessing perfect matchings must have an even number of vertices. Therefore throughout this section we assume that is even.

It is well known that if a tree has a perfect matching, then this perfect matching is unique: namely, a pendent vertex has to be matched with its unique neighbor  , and then forms the perfect matching of  .

Let be a -starlike tree consisting of a central vertex  , a pendent vertex, a pendent path of length  , and pendant paths of length  , all attached to  .

Theorem 4.1

The tree has maximum eccentric connectivity index among trees with perfect matching and maximum vertex degree  .

Proof: Let be an arbitrary tree with perfect matching and let be a vertex of degree  , with neighbors  . Let be the maximal subtrees rooted at  , respectively. Then at most one of the numbers can be odd (if and have odd number of vertices, then their roots and will be unmatched). Since the number of vertices of is even, there exists exactly one among with odd number of vertices.

Using Theorem 2.1, we may transform each into a path attached to – while simultaneously decreasing and keeping the existence of a perfect matching. Assume that has odd number of vertices, while the remaining trees have even number of vertices. We apply a transformation similar to the one in Theorem 2.1, but instead of moving one vertex, we move two vertices in order to keep the existence of a perfect matching. Thus, if then

Using this transformation we may reduce to one vertex, the trees to two vertices, leaving with vertices, and thus obtaining  . Since all times we strictly decreased  , we conclude that has minimum eccentric connectivity index among the trees with perfect matching and maximum vertex degree  .


The path has maximum, while has minimum eccentric connectivity index among trees with perfect matchings.

5 The minimum eccentric connectivity index of trees
with fixed number of pendent vertices

In [22] the authors determinate the -vertex trees with pendent vertices,  , with the maximum eccentric connectivity index, and, consecutively, the extremal trees with the maximum, second-maximum and third-maximum eccentric connectivity index for  . For the completeness, here we determine the -vertex trees with pendent vertices that have minimum eccentric connectivity index.

Definition 5.1

Let be a vertex of a tree of degree  . Suppose that are pendent paths incident with  , with lengths  . Let be the neighbor of distinct from the starting vertices of paths  , respectively. We form a tree by removing the edges from and adding new edges incident with  . We say that is a -transform of and write  .

Theorem 5.1

Let be a -transform of a tree of order  . Let be a non-central vertex, furthest from the root among all branching vertices (with degree greater than ). Then

Proof: The degrees of vertices and have changed – namely,  . Since the furthest vertex from does not belong to and for  , it follows that the eccentricities of all vertices different from do not change after transformation. The eccentricities of vertices from also remain the same, while the eccentricities of vertices from decrease by one. Using the equality  , it follows that

This completes the proof.


The -starlike tree is balanced if all paths have almost equal lengths, i.e., for every  .

Theorem 5.2

The balanced -starlike tree has minimum eccentric connectivity index among trees with pendent vertices,  .

Proof: Let be a rooted -vertex tree with pendent vertices. If contains only one vertex of degree greater than two, we can apply Theorem 2.1 in order to arrive at the balanced starlike tree  , without changing the number of pendent vertices. If has several vertices of degree greater than  , such that there are only pendent paths attached below them, then we take the one most distant from the center vertex of  . By repetitive application of the transformation and balancing pendant paths, the eccentric connectivity index decreases.

Assume that we arrived at a tree with two centers with only pendent paths attached at both centers. If all pendent paths have equal lengths, then  . Since we can reattach pendent paths at any central vertex without changing  , it follows that there are exactly extremal trees with minimum eccentric connectivity index in this special case.

Now, let be the path with length attached to and let be the shortest path of length attached to  . After applying the transformation at vertex  , the eccentric connectivity index remains the same. If we apply the transformation from Theorem 2.1 to two pendant paths of lengths and attached at  , we will strictly decrease the eccentric connectivity index. Finally, we conclude that is the unique extremal tree that minimizes among -vertex trees with pendent vertices for  .


6 Chemical trees with minimal eccentric connectivity index

Theorem 6.1

Let be a rooted tree, with a center vertex as root. Let be the vertex closest to the root vertex, such that  . Let be the pendent vertex most distant from the root, adjacent to vertex  , such that  . Construct a tree by deleting the edge and inserting the new edge  . Then

Proof: In the transformation the degrees of vertices other than and remain the same, while and  . Since the tree is rooted at the center vertex, the radius of is equal to  . Furthermore, there exists a vertex in a different subtree attached to the center vertex, such that or  . From the condition  , it follows that and  .

By rotating the edge to  , the eccentricity of vertices other than decrease if and only if is the only vertex at distance from the center vertex. Otherwise, the eccentricities remain the same. In both cases, we have

This completes the proof.


The Volkmann tree is a tree on vertices and maximum vertex degree  , defined as follows [3, 4]. Start with the root having children. Every vertex different from the root, which is not in one of the last two levels, has exactly children. In the last level, while not all vertices need to exist, the vertices that do exist fill the level consecutively. Thus, at most one vertex on the level second to last has its degree different from and  . For more details on Volkmann trees see [3, 4, 6]. In [3, 4] it was shown that among trees with fixed and  , the Volkmann tree has minimum Wiener index. Volkmann trees have also other extremal properties among trees with fixed and [6, 8, 15, 20].

Figure 2: The Volkmann tree  .
Theorem 6.2

Let be an arbitrary tree on vertices with maximum vertex degree  . Then

Proof: Among -vertex trees with maximum degree  , let be the extremal tree with minimum eccentric connectivity index. Assume that is a vertex closest to the root vertex  , with and let be the pendent vertex most distant from the root, adjacent to vertex  . Also, let be the greatest integer, such that

First, we will show that the radius of has to be less than or equal to  . Assume that  . Since the distance from the center vertex to is less than or equal to  , it follows that

If strict inequality holds, then we can apply Theorem 6.1 and decrease the eccentric connectivity index – which contradicts to the assumption that is the tree with minimum  . Therefore, and after performing the transformation from Theorem 6.1, the eccentric connectivity index does not change. According to the definition of the number  , after finitely many transformations, the vertex  will be the only vertex at distance from the center vertex and we will strictly decrease  . Also, this means that for the case  , the Volkmann tree is the unique tree with minimum eccentric connectivity index.

Now, we can assume that the radius of is equal  . If the distance is less than  , it follows again that  , which is impossible. Therefore, the levels are full (level contains exactly vertices), while the -th and -th levels contain

vertices.

Assume that has only one center vertex – then and  . If  , we can apply the transformation from Theorem 6.1 and strictly decrease  . Thus, for  , the -th level is also full and the pendent vertices in the -th level can be arbitrarily assigned. Using the same argument, for  , the extremal trees are bicentral. By completing the -th level, we do not change the eccentric connectivity index – since  . Finally, and the result follows.


In Table 1 we give the minimum value of eccentric connectivity index among vertex trees with maximum vertex degree  , together with the number of such extremal trees (of which one is the Volkman tree). Note that for the number of extremal trees is  , and for holds  .

7 A linear algorithm for calculating the eccentric connectivity index of a tree

Let be a rooted tree, with a center vertex as root. Let be the neighbors of the center vertex  , and be the corresponding rooted subtrees. Let be the length of the longest path from in the subtree  ,  .

Lemma 7.1

The eccentricity of the vertex equals

11 150 ; 1 79 ; 3 62 ; 5 60 ; 6 49 ; 1 49 ; 1 49 ; 1 49 ; 1 30 ; 1
12 182 ; 1 88 ; 3 69 ; 4 67 ; 8 54 ; 1 54 ; 1 54 ; 1 54 ; 1 54 ; 1
13 216 ; 1 97 ; 1 76 ; 4 74 ; 9 72 ; 10 59 ; 1 59 ; 1 59 ; 1 59 ; 1
14 254 ; 1 106 ; 1 83 ; 3 81 ; 11 79 ; 12 64 ; 1 64 ; 1 64 ; 1 64 ; 1
15 294 ; 1 130 ; 7 90 ; 2 88 ; 11 86 ; 16 84 ; 14 69 ; 1 69 ; 1 69 ; 1
16 338 ; 1 141 ; 10 97 ; 1 95 ; 12 93 ; 19 91 ; 19 74 ; 1 74 ; 1 74 ; 1
17 384 ; 1 152 ; 7 104 ; 1 102 ; 11 100 ; 23 98 ; 24 96 ; 21 79 ; 1 79 ; 1
18 434 ; 1 163 ; 7 138 ; 24 109 ; 11 107 ; 25 105 ; 31 103 ; 27 84 ; 1 84 ; 1
19 486 ; 1 174 ; 4 147 ; 20 116 ; 9 114 ; 29 112 ; 37 110 ; 36 108 ; 29 89 ; 1
20 542 ; 1 185 ; 3 156 ; 18 123 ; 8 121 ; 30 119 ; 46 117 ; 45 115 ; 39 94 ; 1
11
12 33 ; 1
13 59 ; 1 36 ; 1
14 64 ; 1 64 ; 1 39 ; 1
15 69 ; 1 69 ; 1 69 ; 1 42 ; 1
16 74 ; 1 74 ; 1 74 ; 1 74 ; 1 45 ; 1
17 79 ; 1 79 ; 1 79 ; 1 79 ; 1 79 ; 1 48 ; 1
18 84 ; 1 84 ; 1 84 ; 1 84 ; 1 84 ; 1 84 ; 1 51 ; 1
19 89 ; 1 89 ; 1 89 ; 1 89 ; 1 89 ; 1 89 ; 1 89 ; 1 54 ; 1
20 94 ; 1 94 ; 1 94 ; 1 94 ; 1 94 ; 1 94 ; 1 94 ; 1 94 ; 1 57 ; 1

Table 1. The minimal value of the eccentricity connectivity index of trees with vertices and maximum vertex degree  , and the number of such extremal trees.

Proof: We show that the longest path starting at vertex has to traverse the center vertex  . This means that the eccentricity of is equal to the sum of and the longest path starting at and not contained in  . Assume that the longest path from stays in the subtree  , and let be the vertex from at the smallest distance from the root  . Then  . Since the root vertex is a center of  , we have and consequently

This means that is strictly greater than  , which is a contradiction.


We now present a simple linear algorithm for calculating the eccentric connectivity index of a tree  . First, find a center vertex of a tree – this can be done in time (see [1] for details). For every vertex  , we have to find the length of the longest path from in the subtree rooted at  . This can be done inductively using depth–first search, also in time  . If represents the length of the longest path in the subtree rooted at  , then

where denotes the parent of vertex in  . For all neighbors of the center vertex  , we can calculate the maximum  . Finally, for every vertex we calculate the eccentricity in using Lemma 7.1, and sum  .

The time complexity of the algorithm is linear  , and the memory used is  , since we need three additional arrays of length  .

Acknowledgement. This work was supported by the research grants 144015G and 144007 of the Serbian Ministry of Science and Technological Development.

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