1 Introduction
A chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory where atoms are represented by vertices and chemical bonds by edges. Arthur Cayley [1]
was probably the first to publish results that consider chemical graphs. In an attempt to analyze the chemical properties of alkanes, Wiener
[11] has introcuced the path number index, nowadays called Wiener index, which is defined as the sum of the lengths of the shortest paths between all pairs of vertices. Mathematical properties and chemical applications of this distancebased index have been widely researched.Numerous other topological indices are used for quantitative structureproperty relationship (QSPR) and quantitative structureactivity relationship (QSAR) studies that help to describe and understand the structure of molecules [10, 6], among which the eccentric connectivity index which can be defined as follows. Let be a simple connected undirected graph. The distance between two vertices and in is the number of edges of a shortest path in connecting and . The eccentricity of a vertex is the maximum distance between and any other vertex, that is . The eccentric connectivity index of is defined by
This index was introduced by Sharma et al. in [9] and successfully used for mathematical models of biological activities of diverse nature [2, 3, 7, 8, 5]. Recently, Hauweele et al.[4] have characterized those graphs which have the largest eccentric connectivity index among all connected graphs of given order . These results are summarized in Table 1, where

is the complete graph of order ;

is the path of order ;

is the wheel of order , i.e., the graph obtained by joining a vertex to all vertices of a cycle of order ;

is the graph obtained from by removing a maximum matching and, if
is odd, an additional edge adjacent to the unique vertex that still degree
; 
is the graph constructed from a path by joining each vertex of a clique to , and .
optimal graphs  

1  
2  
3  and 
4  
5  and 
6  
7  
8  and 
In addition to the abovementioned graphs, we will also consider the following ones:

is the chordless cycle of order ;

is the graph of order obtained by linking all vertices of a stable set of vertices with all vertices of a clique . The graph is called a star.
Also, for and , let be the graph of order obtained by adding a dominating vertex (i.e., a vertex linked to all other vertices) to the graph or order having vertices of degree 0, and

vertices of degree 1 if is odd;

vertices of degree 1 and one vertex of degree 2 if is even.
For illustration, and are drawn on Figure 1. Note that . Moreover, has two dominating vertices while and have exactly one dominating vertex for all and .
In this paper, we first give an alternative proof to a result of Zhou and Du [12] showing that the stars are the only graphs with smallest eccentric connectivity index among all connected graphs of given order . These graphs have pending vertices (i.e., vertices of degree 1). We then consider all pairs of integers with and characterize the graphs with smallest eccentric connectivity index among all connected graphs of order with pending vertices.
2 Minimizing for graphs with fixed order
and are the only connected graphs with 1 and 2 vertices, respectively, while and are the only connected graphs with 3 vertices. Since , all connected graphs of given order have the same eccentric connectivity index. From now on, we therefore only consider connected graphs with fixed order . A proof of the following theorem was already given by Zhou and Du in [12]. Ours is slightly different.
Theorem 1.
Let be a connected graph of order . Then , with equality if and only if .
Proof.
Let be the number of dominating vertices (i.e., vertices of degree ) in . We distinguish three cases.

If , then let be the dominating vertex in . Clearly, and . All vertices have eccentricity , while their degree is at least 1 (since is connected). Hence, , with equality if and only if all have degree 1, i.e., .

If , then all dominating vertices have , while all nondominating vertices have and , which implies . If , we therefore have , while if , we have .

If , then every pending vertex has since its only neighbor is a nondominating vertex. Since the eccentricity of the nonpending vertices is at least two, we have for all vertices in , which implies .
∎
Stars have pending vertices. As will be shown in the next section, a similar result is more challenging when the total number of pending vertices is fixed to a value strictly smaller than .
3 Minimizing for graphs with fixed order and fixed number of pending vertices
Let be a connected graph of order with pending vertices. Clearly, , and if . For , let and be the two nonpending vertices. Note that is adjacent to since is connected. Clearly, is obtained by linking vertices of a stable set of vertices to , and the other vertices of to . The pending vertices have and , while and . Hence for all graphs of order with pending vertices.
The above observations show that all graphs of order with a fixed number of pending vertices have the same eccentric connectivity index. As will be shown, this is not the case when and . We will prove that is almost always the unique graph minimizing the eccentric connectivity index. Note that
Theorem 2.
Let be a connected graph of order with pending vertices and one dominating vertex. Then , with equality if and only if .
Proof.
The dominating vertex in has , the pending vertices have , and the other vertices have and . Hence, is minimized if all nonpending and nondominating vertices have degree 2, except one that has degree 3 if is odd. In other words, is minimized if and only if . ∎
Theorem 3.
Let be a connected graph of order , with at least two dominating vertices.

If then , with equality if and only if .

If then , with equality if and only if or .

If then , with equality if and only if .
Proof.
Let be the number of dominating vertices in . Then for all dominating vertices , while and for all other vertices . Hence, .

If then . Since , , and , we conclude that , with equality if and only if , which is the case when .

If then . Since , and , we conclude that , with equality if and only if or , which is the case when or .

If then is minimized for , which is the case when .
∎
Theorem 4.
Let be a connected graph of order , with pending vertices and no dominating vertex. Then unless , and , in which case .
Proof.
Let be the subset of vertices in such that . If is empty, then all nonpending vertices in have and (since has no dominating vertex), and at least one of these two inequalities is strict, which implies . Also, every pending vertex has since their only neighbor is not dominant. Hence, . Since , we have .
So, assume . Let be a vertex in , and let be its two neighbors. Also, let , , and . Since , all vertices of belong to . We finally define as the subset of that contains all vertices of with (i.e., their only neighbors are and ).
Case 1: is adjacent to .
else is a dominating vertex, and else is dominating. Let be the graph obtained from by replacing every edge linking to a vertex with an edge linking to , and by removing all edges linking to a vertex of . Clearly, is also a connected graph of order with pending vertices, and is the only dominating vertex in . It follows from Theorem 2 that . Also,

and ;

and for all ;

and for all ;

and for all .
Hence,
Moreover,

;

We therefore have
This implies .
Case 2: is not adjacent to , and both and are nonempty.
Let be the graph obtained from by adding an edge linking to , by replacing every edge linking to a vertex with an edge linking to , and by removing all edges linking to a vertex of . Clearly, is also a connected graph of order with pending vertices. As in the previous case, we have
Moreover, and , while and , which implies

;

We therefore have
If , is the only dominating vertex in , and . It then follows from Theorem 2 that . So assume . Since , and , we have and . Hence, once again, is the only dominating vertex in , and we know from Theorem 2 that .

If , or , then .

If and , there are two possible cases:

if the vertex in is not adjacent to the vertex in , then , , and . Hence, ;

if the vertex in is adjacent to the vertex in , then , , and . Hence, ;

Case 3: is not adjacent to , and at least one of and is empty.
Without loss of generality, suppose . We distinguish two subcases.
Case 3.1: .
Since , . Also, since , there is a nonpending vertex . Let be the graph obtained from by removing the edge linking and and by linking to and to . Note that is a connected graph of order with pending vertices : while was pending in , but not , the situation is the opposite in . Note also that Theorem 2 implies since is the only dominating vertex in . We then have:

, and , which gives ;

, and , which gives ;

, and , which gives ;

, and , which gives ;

and for all . Since has a neighbor in of degree at least 2, we have .
Hence, , which implies .
Case 3.2: .
Let be the vertices in . Remember that the unique neighbors of these vertices are and . Let be the graph obtained from as follows. We first add an edge linking to . Then, for every odd , we add an edge linking to and remove the edges linking to and to . We then have

and for all ;

, , , and ;

, , , and .
Hence,
IF or , then , and since is then the only dominating vertex in , we know from Theorem 2 that . So, assume and :

if then , , and which implies ;

if then , , and which implies .
∎
We can now combine these results as follows. Assume is a connected graph of order with pending vertices. If , then has at most one dominating vertex, and it follows from Theorems 2 and 4 that is the only graph with maximum eccentric connectivity index. If and , then cannot contain exactly one dominating vertex, and Theorems 3 and 4 show that is the only graph with maximum eccentric connectivity index. If and , Theorems 2, 3 and 4 show that , , and are the only candidates to minimize the eccentric connectivity index, and since , the four graphs are the optimal ones. If and then we know from Theorems 2, 3 and 4 that and are the only candidates to minimize the eccentric connectivity index. Since , and for , we deduce that is the only graph with maximum eccentric connectivity index when and , while is the only optimal graph when and . This is summarized in the following Corollary.
Corollary 5.
Let be a connected graph of order with pending vertices.

If then , with equality if and only if ;

If then

if then , with equality if and only if ;

if then , with equality if and only if , , or ;

if then , with equality if and only if ;

if then , with equality if and only if .

4 Conclusion
We have characterized the graphs with smallest eccentric connectivity index among those of fixed order and fixed or nonfixed number of pending vertices. Such a characterization for graphs with a fixed order and a fixed size was given in [12]. It reads as follows.
Theorem 6.
Let be a connected graph of order with edges, where Also, let
Then , with equality if and only if has dominating vertices and vertices of eccentricity 2.
It is, however, an open question to characterize the graphs with largest eccentric connectivity index among those of fixed order and fixed size . The following conjecture appears in [4], where is the graph of order constructed from a path by joining each vertex of a clique to and , and vertices of the clique to .
Conjecture 7.
Let be a connected graph of order with edges, where Also, let
Then , with equality if and only if or , and is the graph constructed from a path , by joining vertices of a clique to and the other vertices of to .
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