Bounds on the edge-Wiener index of cactiwith n vertices and t cycles

# Bounds on the edge-Wiener index of cacti with n vertices and t cycles

Siyan Liu, Rong-Xia Hao***Corresponding author. Email: 14275011@bjtu.edu.cn (Siyan Liu), rxhao@bjtu.edu.cn (Rong-Xia Hao), he1046436120@126.com (Shengjie He), Shengjie He
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China

The edge-Wiener index of a connected graph is the sum of distances between all pairs of edges of . A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. Let denote the class of all cacti with vertices and cycles. In this paper, the upper bound and lower bound on the edge-Wiener index of graphs in are identified and the corresponding extremal graphs are characterized.

Keywords: Cactus; Edge-Wiener index; Upper bound; Lower bound.

## 1 Introduction

Throughout this paper, all graphs we considered are finite, undirected, and simple. Let be a connected graph with vertex set and edge set . For a vertex , the of , denote by , is the number of vertices which are adjacent to . For a vertex , denote by , the set of the vertices which are adjacent to . Call a vertex a of , if and call an edge a of , if or . By and we denote the graph obtained from by deleting a vertex , or an edge , respectively (This notation is naturally extended if more than one vertex or edge are deleted). Similarly, is obtained from by adding an edge . For any two vertices , let denote the distance between and in . Denote by , and a path, star and cycle on vertices, respectively. We refer to Bondy and Murty [2] for notation and terminologies used but not defined here.

The Wiener index is one of the oldest and the most thoroughly studied topological indices. The Wiener index of a graph is defined as

 W(G)=∑{u,v}⊆V(G)dG(u,v).

The edge-Wiener index is defined as the sum of distances between all pairs of edges, namely as

 We(G)=∑{f,g}⊆E(G)dG(f,g).

where denotes the distance between and in , and also the distance between the corresponding vertices in the line graph of . Note that for any two distinct edges and in , the distance between and equals . In the case, when and coincide, we have . Nowadays, the Wiener index is a well-known and much studied graph invariant e.g., [4, 6, 9, 10, 11, 12]. In  [3], Dankelmanna et al. gave bounds on in terms of order and size. But there is not so many conclusions about edge-Wiener index. So we pay attention to it and hope to get more conclusions.

A connected graph is said to be a if any two of its cycles have at most one common vertex. Let be the set of all -vertex cacti, each containing exactly cycles. A graph is a if each block has at most two cut vertices and each cut vertex is shared by exactly two blocks. Obviously, any chain cactus with at least two blocks contains exactly two blocks that have only one cut-vertex. Such blocks are called blocks. If all the cycles in a have exactly one common vertex, then they form a . In  [5], the Wiener index of cacti with vertices and cycles was studied by Gutman. Wang  [13, 14] determined the the lower bounds on Szeged index and revised Szeged index of cacti with vertices and cycles. He et al. [7] determined the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order and cycles.

In this paper, by using the methods similar to Gutman [5], the edge-Wiener index of the cacti with vertices and cycles is studied. Moreover, the lower bound on edge-Wiener index of the cacti with given cycles is determined and the corresponding extremal graph is identified. Furthermore, the upper bound on edge-Wiener index of the cacti with given cycles and the corresponding extremal graph are established as well.

## 2 Preliminaries

In this section, we give some preliminary results which will be used in the subsequent sections.

###### Lemma 1.

[9]) The Wiener index of and is

 W(Pn)=16n(n+1)(n−1),
 W(Cn)=⎧⎨⎩18n(n2−1),if n is odd % ;18n3,if n is even.
###### Lemma 2.

Let and be two connected graphs with disjoint vertex sets where , . Let , for , . Construct the graph by identifying the vertices and , and denote the new vertex by (see Figure 1). Then

 We(G)=We(G1)+We(G2)+m1∑f∈E(G2)dG(f,u)+m2∑g∈E(G1)dG(g,u)+m1m2.

Proof.   According to the definition of edge-Wiener index, we have

 We(G)=∑f,g∈E(G)dG(f,g)=∑f,g∈E(G1)dG1(f,g)+∑f,g∈E(G2)dG2(f,g)+∑f∈E(G1),g∈E(G2)dG(f,g)=We(G1)+We(G2)+∑f∈E(G1),g∈E(G2)[dG(f,u)+dG(g,u)+1]=We(G1)+We(G2)+∑f∈E(G1),g∈E(G2)dG(f,u)+∑f∈E(G1),g∈E(G2)dG(g,u)+m1m2=We(G1)+We(G2)+m1∑f∈E(G2)dG(f,u)+m2∑g∈E(G1)dG(g,u)+m1m2.

That’s the end of the proof. ∎

###### Lemma 3.

Let be a graph with a cut edge , and be the graph obtained from by contracting the edge and adding a pendant edge attaching at the contracting vertex; see Figure 2. Let for . If for , we have .

Proof.   We calculate the edge-Wiener index of and , respectively.

 We(G)=∑f,g∈E(G)dG(f,g)=∑f,g∈E(G1)dG1(f,g)+∑f,g∈E(G2)dG2(f,g)+∑f∈E(G1),g∈E(G2)dG(f,g)+∑f∈E(G1)dG(f,v1v2)+∑g∈E(G2)dG(g,v1v2)=We(G1)+We(G2)+∑f∈E(G1),g∈E(G2)[dG(f,v1)+dG(g,v2)+1+1]+∑f∈E(G1)dG(f,v1v2)+∑g∈E(G2)dG(g,v1v2).

Also, use the same method, then we have

 We(G′)=∑f,g∈E(G′)dG′(f,g)=We(G′1)+We(G′2)+∑f∈E(G′1),g∈E(G′2)[dG′(f,v1)+dG′(g,v2)+1]+∑f∈E(G′1)dG′(f,v1v2)+∑g∈E(G′2)dG′(g,v1v2).

So

 We(G)−We(G′)=∑f∈E(G1),g∈E(G2)1=m1m2>0.

That’s the end of the proof. ∎

###### Lemma 4.

Let be a graph with an even cycle such that has exactly components. Let be the component of that contains and for . Let

 G′=G−2k∑i=2∑w∈NGi(vi)wvi+2k∑i=2∑w∈NGi(vi)wv1.

(see Figure 3). Then we have with equality if and only if is an end-block, that is, .

Proof.   If is an end-block, the lemma holds clearly. Then, one can assume that is not an end-block in the following.

By the definition of edge-Wiener, we can suppose that , which are

 T1=2k∑i=1∑f,g∈E(Hi)[dG(f,g)−dG′(f,g)],T2=∑f,g∈E(C2k)[dG(f,g)−dG′(f,g)],T3=∑1≤i

According to the structure of and , we have clearly. Then we calculate and .

 T3=∑1≤i

By when , we have . Now let’s start . We say the sum of the edges in each branch is equal to , each edge of the cycle is for . The sum distance between and () is equal to .

In graph , , and in graph , , for is the cut vertex of . Minus equal parts, we have

 T4=2k∑i=1∑f∈E(Hi),g∈E(C2k)[dG(f,g)−dG′(f,g)]=∑1≤i≤2k,1≤j≤2k,g∈E(Hj)dG(ei,g)−∑1≤i≤2k,1≤j≤2k,g∈E(Hj)dG′(ei,g).

Now use the previous results,

 T4=∑1≤i≤2k,1≤j≤2kmjdG(ei,vj)−∑1≤i≤2k,1≤j≤2kmjdG′(ei,v1)=∑1≤j≤2k[mj2k∑i=1dG(ei,vj)]−∑1≤j≤2k[mj2k∑i=1dG′(ei,v1)].

In graph , the sum distance between the edge and vertex is and

 2k∑i=1dG(ei,v1)=[(0+1+2+3+⋯k−1)×2]=k(k−1).

that means the sum distance of every edge from to is . Now let’s calculate graph , the cycle has symmetry, for every cut vertex and , we have , so

 T4=[m1k(k−1)+m2k(k−1)+m3k(k−1)+⋯+m2kk(k−1)]−[m1k(k−1)+m2k(k−1)+m3k(k−1)+⋯+m2kk(k−1)]=0.

Now we know , so

 We(G)−We(G′)=T1+T2+T3+T4=T3>0.

That’s the end of the proof. ∎

###### Lemma 5.

Let be a graph with an odd cycle such that has exactly components. Let be the component of that contains and for . Let

 G′=G−2k+1∑i=2∑w∈NGi(vi)wvi+2k+1∑i=2∑w∈NGi(vi)wv1.

Then we have with equality if and only if is an end-block, that is, .

Proof.   If is an end-block, the lemma holds clearly. Then, one can assume that is not an end-block in the following.

By the definition of edge-Wiener, we can suppose that , where

 S1=2k+1∑i=1∑f,g∈E(Hi)[dG(f,g)−dG′(f,g)],S2=∑f,g∈E(C2k+1)[dG(f,g)−dG′(f,g)],S3=∑1≤i

According to the structure of and , we have clearly. Then we calculate and .

 S3=∑1≤i

If , . So .

 S4=2k+1∑i=1∑f∈E(Hi),g∈E(C2k+1)[dG(f,g)−dG′(f,g)]=∑1≤i≤2k+1,1≤j≤2k+1,g∈E(Hj)dG(ei,g)−∑1≤i≤2k+1,1≤j≤2k+1,g∈E(Hj)dG′(ei,g)=∑1≤i≤2k+1,1≤j≤2k+1mjdG(ei,vj)−∑1≤i≤2k+1,1≤j≤2k+1mjdG′(ei,v1)=∑1≤j≤2k+1[mj2k+1∑i=1dG(ei,vj)]−∑1≤j≤2k+1[mj2k+1∑i=1dG′(ei,v1)].

It can be checked that

 2k+1∑i=1dG(ei,v1)=[(0+1+2+3+⋯+k−1)×2+k]=k2.

Then,

 S4=[m1k2+m2k2+m3k2+⋯+m2kk2+m2k+1k2]−[m1k2+m2k2+m3k2+⋯+m2kk2+m2k+1k2]=0.

Now we know , so

 We(G)−We(G′)=S1+S2+S3+S4=S3>0

That’s the end of the proof. ∎

###### Lemma 6.

Let be an end-block of with . Let ; see Figure 4. Let . Then we have .

Proof.   We calculate the edge-Wiener index of and , respectively. We will deal with the problem with two cases according to the parity of .

Case 1. is even.

By the definition of edge-Wiener, we can suppose that

 We(G)−We(G′)=T1+T2+T3+T4+T5+T6.

Where,

 T1=∑f,g∈E(G0)dG(f,g)−∑f,g∈E(G0)dG′(f,g),T2=∑f,g∈E(Cr)dG(f,g)−∑f,g∈E(Cr−2)dG′(f,g),T3=∑f∈E(Cr),g∈E(G0)dG(f,g)−∑f∈E(Cr−2),g∈E(G0)dG′(f,g),T4=0−2∑f∈E(Cr−2)dG′(v1v2,f),T5=0−2∑f∈E(G0)dG′(v1v2,f),T6=0−dG′(v1vr,v1v2)=−1.

It is not difficult to find that and the edge-Wiener index of cycle is equal to the Wiener index, so as to . By the fact that the Wiener index of a cycle equal to its edge-Wiener, then by Lemma 1, one has that

 T2=∑f,g∈E(Cr)dG(f,g)−∑f,g∈E(Cr−2)dG′(f,g)=We(Cr)−We(Cr−2)=38r3−38(r−3)3.

By the symmetry of the cycle, we have

 T3=∑f∈E(Cr),g∈E(G0)dG(f,g)−∑f∈E(Cr−2),g∈E(G0)dG′(f,g)=∑f∈E(Cr−2),g∈E(G0)dG′(f,g)+2∑f∈E(G0)dG′(f,e0)−∑f∈E(Cr−2),g∈E(G0)dG′(f,g)=2∑f∈E(G0)dG′(f,e0)

with . By calculation, we have

 T4=0−2∑f∈E(Cr−2)dG′(v1v2,f)=−2×2×(1+2+3+⋯+r2−1)=−4r2−1∑i=1i,
 T5=−2∑f∈E(G0)dG′(v1v2,f),T6=−1.

Thus,

 We(G)−We(G′)=T1+T2+T3+T4+T5+T6=94r2−92r+3+2∑f∈E(G0)(r2−1)−12(r2−2r)−1=74r2+(m−72)r−2m+2(r≥5).

We can get it easily that for is even when .

Case 2. is odd.

By the definition of edge-Wiener, we can suppose that

 We(G)−We(G′)=S1+S2+S3+S4+S5+S6.

Where,

 S1=∑f,g∈E(G0)dG(f,g)−∑f,g∈E(G′0)dG′(f,g),S2=∑f,g∈E(Cr)dG(f,g)−∑f,g∈E(Cr−2)dG′(f,g),S3=∑f∈E(Cr),g∈E(G0)dG(f,g)−∑f∈E(Cr−2),g∈E(G0)dG′(f,g),S4=0−2∑f∈E(Cr−2)dG′(v1v2,f),S5=0−2∑f∈E(G0)dG′(v1v2,f),S6=0−dG′(v1vr,v1v2)=−1.

It can be checked that , and ,

 S3=2∑f∈E(G0)[dG′(f,v0)+(r−12−1)+1]+∑f∈E(G0)(r−12−r−32),
 S4=0−2∑f∈E(Cr−2)dG′(v1v2,f)=−[2×(2r−32∑i=1i+r−32+1)].

Then,

 We(G)−We(G′)=S1+S2+S3+S4+S5+S6=14r2+(m−12)r−2m+34.

We can get it easily that