# Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

Authors: Zhaoyang Luo; Jianliang Wu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

LetG be a connected graph. The first and second Zagreb eccentricity indices of G are defined as M
1
*
(
G
)
=
∑
v
∈
V
(
G
)
‍
ε
G
2
(
v
) and M
2
*
(
G
)
=
∑
u
v
∈
E
(
G
)
‍
ε
G
(
u
)
ε
G
(
v
), where ε
G
(
v
) is the eccentricity of the vertex v in G and ε
G
2
(
v
)
=
(
ε
G
(
v
)
)
2. Suppose that G
(
U
)
⊓
H
(
∅
≠
U
⊆
V
(
G
)
) is the generalized hierarchical product of two connected graphs G and H. In this paper, the Zagreb eccentricity indices M
1
* and M
2
* of G
(
U
)
⊓
H are computed. Moreover, we present explicit formulas for the M
1
* and M
2
* of S-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.


---

## Body

## 1. Introduction

A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity, which is used to understand properties of chemical compounds in theoretical chemistry [1].Up to now, hundreds of topological indices have been defined in chemical literatures, various applications of these topological indices have been found, and many mathematical properties are also investigated. Wiener indexW is the first topological index, introduced by American chemist Wiener, for investigating boiling points of alkanes in 1947 [2]. The well known degree-based topological indices are the first and second Zagreb indices M
1 and M
2, which have been introduced by Gutman and Trinajstić [3] and applied to study molecular chirality in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) analysis and so forth. Resently, the first and second Zagreb eccentricity indices M
1
* and M
2
* have been introduced by Ghorbani and Hosseinzadeh [4] and Vukičević and Graovac [5] as the revised version of the Zagreb indices M
1 and M
2, respectively. They computed the Zagreb eccentricity indices of some composite graphs and showed that M
1
*
(
G
)
/
|
G
|
≥
M
2
*
(
G
)
/
|
E
(
G
)
| holds for all acyclic and unicyclic graphs and that neither this nor the opposite inequality holds for all bicyclic graphs. For further results of the Zagreb eccentricity indices, we encourage the reader to refer to [6–8].In 2009, Spain mathematicians Barrière and coauthors [9] introduced a new composite graph, namely, hierarchical product graph. In the same year, this team also reported a generalization of both Cartesian and the hierarchical product of graphs, namely, the generalized hierarchical product of graphs in [10]. After that, many results for some topological indices of the (generalized) hierarchical product of graphs are reported; see [11–17].In this paper, the Zagreb eccentricity indices of the generalized hierarchical product graphG
(
U
)
⊓
H are computed and as some special cases of G
(
U
)
⊓
H, the Zagreb eccentricity indices of the Cartesian product graph G
□
H, the S-sum graph G
+
S
H, and the cluster product graph G
{
H
} are determined, respectively. Moreover, as applications, we present explicit formulas for the M
1
* and M
2
* indices of the C
4 nanotorus C
m
□
C
n, the C
4 nanotubes P
m
□
C
n, the zig-zay polyhex nanotube T
U
H
C
6
[
2
n
,
2
], the hexagonal chain L
n, and so forth.


## 2. Preliminaries

Throughout this paper, all graphs are simple, finite, and undirected. For terminology and notations that are not defined here, we refer the reader to West [18].LetG
=
(
V
(
G
)
,
E
(
G
)
,
ψ
G
) be a graph with the vertex set V
(
G
)
≠
∅, the edge set E
(
G
), and an incidence function ψ
G that associates with each edge of G, an unordered pair of vertices of G. If e is an edge and u and v are vertices such that ψ
G
(
e
)
=
u
v, then e is said to join u and v, and the vertices u and v are called the ends of e. The cardinality of V
(
G
) and E
(
G
) is denoted by |
G
| and |
E
(
G
)
|, respectively. We denote the degree and the neighborhood of a vertex v of G by d
G
(
v
) and N
G
(
v
); then d
G
(
v
)
=
|
N
G
(
v
)
|. As usual, the distance between vertices u and v of a connected graph G, denoted by d
G
(
u
,
v
), is defined as the number of edges in a shortest path connecting the vertices u and v. Suppose that d
(
x
∣
G
)
=
∑
v
∈
V
(
G
)
‍
d
G
(
x
,
v
) and d
2
(
x
∣
G
)
=
∑
v
∈
V
(
G
)
‍
(
d
G
(
x
,
v
)
)
2. The eccentricity ε
G
(
v
) of a vertex v in G is the largest distance between v and any other vertex u of G; that is, ε
G
(
v
)
=
max
⁡
u
∈
V
(
G
)
d
G
(
u
,
v
). For two graphs G and H, if there exist two bijections θ
:
V
(
G
)
→
V
(
H
) and φ
:
E
(
G
)
→
E
(
H
) such that ψ
G
(
e
)
=
u
v if and only if ψ
H
(
φ
(
e
)
)
=
θ
(
u
)
θ
(
v
), then we say that G and H are isomorphic, denoted by G
≅
H. Let Top
(
G
) denote a certain topological index of G. In general, if G
≅
H, then Top
(
G
)
=
Top
(
H
).The total eccentricity and the eccentric connectivity indicesζ
(
G
) and ξ
c
(
G
) of graph G are defined as ζ
(
G
)
=
∑
v
∈
V
(
G
)
‍
ε
G
(
v
) and ξ
c
(
G
)
=
∑
u
v
∈
E
(
G
)
‍
[
ε
G
(
u
)
+
ε
G
(
v
)
]
=
∑
v
∈
V
(
G
)
‍
d
G
(
v
)
ε
G
(
v
), respectively. The Zagreb indices of G are defined as M
1
(
G
)
=
∑
u
v
∈
E
(
G
)
‍
[
d
G
(
u
)
+
d
G
(
v
)
]
=
∑
u
∈
V
(
G
)
‍
(
d
G
(
u
)
)
2 and M
2
(
G
)
=
∑
u
v
∈
E
(
G
)
‍
d
G
(
u
)
d
G
(
v
). Very recently, the topological indices based on vertex eccentricities attracted some attention in chemistry. In an analogy with the Zagreb indices, the first and second Zagreb eccentricity indices M
1
* and M
2
* of a connected graph G are defined by [4, 5]. That is,
(1)
M
1
*
(
G
)
=
∑
v
∈
V
(
G
)
‍
ε
G
2
(
v
)
=
∑
v
∈
V
(
G
)
‍
(
ε
G
(
v
)
)
2
,
M
2
*
(
G
)
=
∑
u
v
∈
E
(
G
)
‍
ε
G
(
u
)
ε
G
(
v
)
.Lemma 1 (see [4]).
LetK
n be the complete graph of order n; then ζ
(
K
n
)
=
n, ξ
c
(
K
n
)
=
n
(
n
-
1
), M
1
*
(
K
n
)
=
n, and M
2
*
(
K
n
)
=
(
n
2
).Lemma 2 (see [4]).
LetC
n be the cycle of length n; then ζ
(
C
n
)
=
n
⌊
n
/
2
⌋, ξ
c
(
C
n
)
=
2
n
⌊
n
/
2
⌋, and M
1
*
(
C
n
)
=
M
2
*
(
C
n
)
=
n
⌊
n
/
2
⌋
2.Lemma 3 (see [4]).
LetP
n be the path on n
≥
2 vertices. Then
(2)
ζ
(
P
n
)
=
{
1
4
n
(
3
n
-
2
)
,
2
∣
n
,
1
4
(
n
-
1
)
(
3
n
+
1
)
,
2
∤
n
,
(3)
M
1
*
(
P
n
)
=
{
1
12
n
(
n
-
1
)
(
7
n
-
2
)
,
2
∣
n
,
1
12
(
n
-
1
)
(
7
n
2
-
2
n
-
3
)
,
2
∤
n
,
(4)
ξ
c
(
P
n
)
=
{
1
2
(
3
n
2
-
6
n
+
4
)
,
2
∣
n
,
3
2
(
n
-
1
)
2
,
2
∤
n
,
(5)
M
2
*
(
P
n
)
=
{
1
12
n
(
7
n
2
-
21
n
+
20
)
,
2
∣
n
,
1
12
(
n
-
1
)
(
7
n
2
-
14
n
+
3
)
,
2
∤
n
.


## 3. Zagreb Eccentricity Indices of Generalized Hierarchical Product Graphs

In this section, we calculate the Zagreb eccentricity indices of the generalized hierarchical product graphs.Definition 4 (see [9]).
LetG and H be two connected graphs; ∅
≠
U
⊆
V
(
G
). Then the generalized hierarchical product G
(
U
)
⊓
H is the graph with vertex set V
(
G
)
×
V
(
H
) and vertices (
u
r
,
v
i
) and (
u
s
,
v
k
) are adjacent if and only if [
u
r
=
u
s
∈
U and v
i
v
k
∈
E
(
H
)
] or [
v
i
=
v
k
∈
V
(
H
) and u
r
u
s
∈
E
(
G
)
].Given a connected graphG and ∅
≠
U
⊆
V
(
G
), a path connecting vertices x and y through U is a x
U
y-path of G containing some vertex z
∈
U (vertex z could be the vertex x or vertex y). Then the distance through U between x and y is denoted by d
G
(
U
)
(
x
,
y
), which is the length of the shortest path x
U
y in G. Note that if one of the vertex x and y belongs to U, then d
G
(
U
)
(
x
,
y
)
=
d
G
(
x
,
y
); see [13]. Similarly, we define some invariants related to U in G as follows:ε
G
(
U
)
(
v
)
=
max
⁡
u
∈
V
(
G
)
d
G
(
U
)
(
u
,
v
),
ζ
(
G
(
U
)
)
=
∑
v
∈
V
(
G
)
‍
ε
G
(
U
)
(
v
),
ϵ
(
G
(
U
)
)
=
∑
v
∈
U
‍
ε
G
(
U
)
(
v
); see [15],
ξ
c
(
G
(
U
)
)
=
∑
u
v
∈
E
(
G
)
‍
(
ε
G
(
U
)
(
u
)
+
ε
G
(
U
)
(
v
)
); see [15],
M
1
*
(
G
(
U
)
)
=
∑
v
∈
V
(
G
)
‍
ε
G
(
U
)
2
(
v
),
m
1
*
(
G
(
U
)
)
=
∑
v
∈
U
‍
ε
G
(
U
)
2
(
v
),
M
2
*
(
G
(
U
)
)
=
∑
u
v
∈
E
(
G
)
‍
ε
G
(
U
)
(
u
)
ε
G
(
U
)
(
v
).Theorem 5 (see [15]).
Let graphsG and H be connected; ∅
≠
U
⊆
V
(
G
). Then
(6)
ξ
c
(
G
(
U
)
⊓
H
)
=
|
U
|
ξ
c
(
H
)
+
|
H
|
ξ
c
(
G
(
U
)
)
+
2
|
E
(
G
)
|
ζ
(
H
)
+
2
|
E
(
H
)
|
ϵ
(
G
(
U
)
)
.Lemma 6 (see [10]).
LetG and H be two connected graphs and ∅
≠
U
⊆
V
(
G
). Then
(7)
ε
G
(
U
)
⊓
H
(
u
,
v
)
=
ε
G
(
U
)
(
u
)
+
ε
H
(
v
)
.Theorem 7.
Let graphsG and H be connected; ∅
≠
U
⊆
V
(
G
). Then
(i)(8)
M
1
*
(
G
(
U
)
⊓
H
)
=
|
H
|
M
1
*
(
G
(
U
)
)
+
|
G
|
M
1
*
(
H
)
+
2
ζ
(
G
(
U
)
)
ζ
(
H
)
,
(ii)(9)
M
2
*
(
G
(
U
)
⊓
H
)
=
|
E
(
H
)
|
m
1
*
(
G
(
U
)
)
+
|
E
(
G
)
|
M
1
*
(
H
)
+
|
U
|
M
2
*
(
H
)
+
|
H
|
M
2
*
(
G
(
U
)
)
+
ξ
c
(
G
(
U
)
)
ζ
(
H
)
+
ξ
c
(
H
)
ϵ
(
G
(
U
)
)
.Proof.
LetV
(
G
)
=
{
u
1
,
u
2
,
…
,
u
n
} and V
(
H
)
=
{
v
1
,
v
2
,
…
,
v
n
′
}.
(i) By the definition of Zagreb eccentricity indexM
1
* and Lemma 6, we have
(10)
M
1
*
(
G
(
U
)
⊓
H
)
=
∑
(
u
r
,
v
i
)
∈
V
(
G
(
U
)
⊓
H
)
‍
ε
G
(
U
)
⊓
H
2
(
(
u
r
,
v
i
)
)
=
∑
u
r
∈
V
(
G
)
‍
∑
v
i
∈
V
(
H
)
‍
(
ε
G
(
U
)
(
u
r
)
+
ε
H
(
v
i
)
)
2
=
|
H
|
∑
u
r
∈
V
(
G
)
‍
ε
G
(
U
)
2
(
u
r
)
+
|
G
|
∑
v
i
∈
V
(
H
)
‍
ε
H
2
(
v
i
)
+
2
∑
u
r
∈
V
(
G
)
‍
ε
G
(
U
)
(
u
r
)
∑
v
i
∈
V
(
H
)
‍
ε
H
(
v
i
)
=
|
H
|
M
1
*
(
G
(
U
)
)
+
|
G
|
M
1
*
(
H
)
+
2
ζ
(
G
(
U
)
)
ζ
(
H
)
.
(ii) We partition the edges ofG
(
U
)
⊓
H into two subsets E
1 and E
2, as follows:
(11)
E
1
=
{
(
u
r
,
v
i
)
(
u
r
,
v
k
)
∈
E
(
G
(
U
)
⊓
H
)
∣
v
i
v
k
∈
E
(
H
)
,
u
r
∈
U
}
,
E
2
=
{
(
u
r
,
v
i
)
(
u
s
,
v
i
)
∈
E
(
G
(
U
)
⊓
H
)
∣
u
r
u
s
∈
E
(
G
)
,
v
i
∈
V
(
H
)
}
.
From the definition of Zagreb eccentricity index M
2
* and Lemma 6, we get
(12)
M
2
*
(
G
(
U
)
⊓
H
)
=
∑
(
u
r
,
v
i
)
(
u
s
,
v
k
)
∈
E
(
G
(
U
)
⊓
H
)
ε
G
(
U
)
⊓
H
(
(
u
r
,
v
i
)
)
ε
G
(
U
)
⊓
H
(
(
u
s
,
v
k
)
)
=
∑
(
u
r
,
v
i
)
(
u
s
,
v
k
)
∈
E
1
∪
E
2
ε
G
(
U
)
⊓
H
(
(
u
r
,
v
i
)
)
ε
G
(
U
)
⊓
H
(
(
u
s
,
v
k
)
)
=
∑
u
r
∈
U
∑
v
i
v
k
∈
E
(
H
)
[
ε
G
(
U
)
(
u
r
)
+
ε
H
(
v
i
)
]
[
ε
G
(
U
)
(
u
r
)
+
ε
H
(
v
k
)
]
+
∑
v
i
∈
V
(
H
)
∑
u
r
u
s
∈
E
(
G
)
[
ε
G
(
U
)
(
u
r
)
+
ε
H
(
v
i
)
]
×
[
ε
G
(
U
)
(
u
s
)
+
ε
H
(
v
i
)
]
=
|
E
(
H
)
|
∑
u
r
∈
U
ε
G
(
U
)
2
(
u
r
)
+
∑
u
r
∈
U
ε
G
(
U
)
(
u
r
)
×
∑
v
i
v
k
∈
E
(
H
)
(
ε
H
(
v
i
)
+
ε
H
(
v
k
)
)
+
|
U
|
∑
v
i
v
k
∈
E
(
H
)
ε
H
(
v
i
)
ε
H
(
v
k
)
+
|
E
(
G
)
|
∑
v
i
∈
V
(
H
)
ε
H
2
(
v
i
)
+
∑
v
i
∈
V
(
H
)
ε
H
(
v
i
)
∑
u
r
u
s
∈
E
(
G
)
[
ε
G
(
U
)
(
u
r
)
+
ε
G
(
U
)
(
u
s
)
]
+
|
H
|
∑
u
r
u
s
∈
E
(
G
)
ε
G
(
U
)
(
u
r
)
ε
G
(
U
)
(
u
s
)
=
|
E
(
H
)
|
m
1
*
(
G
(
U
)
)
+
ϵ
(
G
(
U
)
)
ξ
c
(
H
)
+
|
U
|
M
2
*
(
H
)
+
|
E
(
G
)
|
M
1
*
(
H
)
+
ζ
(
H
)
ξ
c
(
G
(
U
)
)
+
|
H
|
M
2
*
(
G
(
U
)
)
.
This completes the proofs.Definition 8 (see [13]).
LetG and H be two connected graphs. Then the Cartesian product G
□
H has the vertex set V
(
G
)
×
V
(
H
) and vertices (
u
r
,
v
i
) and (
u
s
,
v
k
) are adjacent if and only if [
u
r
=
u
s
∈
V
(
G
) and v
i
v
k
∈
E
(
H
)
] or [
v
i
=
v
k
∈
V
(
H
) and u
r
u
s
∈
E
(
G
)
], where r
,
s
=
1,2
,
…
,
|
G
| and i
,
k
=
1,2
,
…
,
|
H
|.Note that ifU
=
V
(
G
), then G
(
U
)
⊓
H
≅
G
□
H. So by Theorem 7, the following corollary is obvious.Corollary 9.
LetG and H be two connected graphs. Then
(13)
M
1
*
(
G
□
H
)
=
|
G
|
M
1
*
(
H
)
+
|
H
|
M
1
*
(
G
)
+
2
ζ
(
G
)
ζ
(
H
)
,
(14)
M
2
*
(
G
□
H
)
=
|
E
(
G
)
|
M
1
*
(
H
)
+
|
E
(
H
)
|
M
1
*
(
G
)
+
|
G
|
M
2
*
(
H
)
+
|
H
|
M
2
*
(
G
)
+
ξ
c
(
G
)
ζ
(
H
)
+
ξ
c
(
H
)
ζ
(
G
)
.Remark 10.
Equation (14) corrects the corresponding Corollary  9 in [4]. According to (14), we recompute the second Zagreb eccentricity indices of the C
4 nanotubes as below (see Example 11).Example 11.
Using the results as above, it is easy to obtain the first and second Zagreb eccentricity indices of theC
4 nanotorus C
m
□
C
n
(
m
≥
n
≥
3
) and the C
4 nanotubes P
2
m
□
C
n
(
n
≥
3
) and P
2
m
+
1
□
C
n
(
n
≥
3
). By Corollary 9 and Lemmas 2 and 3, we have(15)
M
1
*
(
C
m
□
C
n
)
=
1
2
M
2
*
(
C
m
□
C
n
)
=
m
n
(
⌊
m
2
⌋
+
⌊
n
2
⌋
)
2
,
M
1
*
(
P
2
m
□
C
n
)
=
m
n
[
14
3
m
2
-
3
m
+
1
3
+
2
⌊
n
2
⌋
2
+
2
(
3
m
-
1
)
⌊
n
2
⌋
]
,
M
1
*
(
P
2
m
+
1
□
C
n
)
=
m
n
(
14
3
m
2
+
4
m
+
1
3
)
+
n
(
2
m
+
1
)
⌊
n
2
⌋
2
+
2
m
n
(
3
m
+
2
)
⌊
n
2
⌋
,
M
2
*
(
P
2
m
□
C
n
)
=
m
n
(
28
3
m
2
-
10
m
+
11
3
)
+
(
4
m
-
1
)
n
⌊
n
2
⌋
2
+
2
(
6
m
2
-
4
m
+
1
)
n
⌊
n
2
⌋
,
M
2
*
(
P
2
m
+
1
□
C
n
)
=
m
n
(
28
3
m
2
+
4
m
-
1
3
)
+
(
4
m
+
1
)
n
⌊
n
2
⌋
2
+
4
m
(
3
m
+
1
)
n
⌊
n
2
⌋
.Lemma 12 (see [19]).
Let□
i
=
1
n
G
i be Cartesian product of n
≥
2 connected graphs G
i. Then(a)
|
□
i
=
1
n
G
i
|
=
|
V
(
□
i
=
1
n
G
i
)
|
=
∏
i
=
1
n
‍
|
G
i
|.
(b)
|
E
(
□
i
=
1
n
G
i
)
|
=
∑
i
=
1
n
‍
[
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|
|
E
(
G
i
)
|
]
=
∑
i
=
1
n
‍
|
E
(
G
i
)
|
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|.
(c)
ζ
(
□
i
=
1
n
G
i
)
=
∑
i
=
1
n
‍
[
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|
ζ
(
G
i
)
]
=
∑
i
=
1
n
‍
ζ
(
G
i
)
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|.Corollary 13.
Let□
i
=
1
n
G
i be Cartesian product of n
≥
2 simple connected graphs G
i. Then
(16)
M
1
*
(
□
i
=
1
n
G
i
)
=
∑
i
=
1
n
‍
[
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|
M
1
*
(
G
i
)
]
+
2
∑
1
≤
s
<
t
≤
n
‍
[
∏
k
=
1
,
k
≠
s
,
t
n
‍
|
G
k
|
ζ
(
G
s
)
ζ
(
G
t
)
]
.Proof.
The casen
=
2 is proved in Corollary 9. We prove the assertion by induction. Suppose the result is valid for n graphs. Then by Lemma 12, we have
(17)
M
1
*
(
□
i
=
1
n
+
1
G
i
)
=
M
1
*
(
G
n
+
1
□
(
□
i
=
1
n
G
i
)
)
=
|
G
n
+
1
|
M
1
*
(
□
i
=
1
n
G
i
)
+
|
□
i
=
1
n
G
i
|
M
1
*
(
G
n
+
1
)
+
2
ζ
(
G
n
+
1
)
ζ
(
□
i
=
1
n
G
i
)
=
|
G
n
+
1
|
∑
i
=
1
n
[
∏
j
=
1
,
j
≠
i
n
|
G
j
|
M
1
*
(
G
i
)
]
+
2
|
G
n
+
1
|
×
∑
1
≤
s
<
t
≤
n
[
∏
k
=
1
,
k
≠
s
,
t
n
|
G
k
|
ζ
(
G
s
)
ζ
(
G
t
)
]
+
∏
i
=
1
n
‍
|
G
i
|
M
1
*
(
G
n
+
1
)
+
2
ζ
(
G
n
+
1
)
∑
i
=
1
n
[
∏
j
=
1
,
j
≠
i
n
|
G
j
|
ζ
(
G
i
)
]
=
∑
i
=
1
n
+
1
[
∏
j
=
1
,
j
≠
i
n
+
1
|
G
j
|
M
1
*
(
G
i
)
]
+
2
∑
1
≤
s
<
t
≤
n
+
1
[
∏
k
=
1
,
k
≠
s
,
t
n
+
1
|
G
k
|
ζ
(
G
s
)
ζ
(
G
t
)
]
.
This completes the proof.Example 14.
The Hamming graphH
n
1
,
n
2
,
…
,
n
s
=
□
i
=
1
n
K
n
i. Thus, by Corollary 13 and Lemma 1, we have M
1
*
(
H
n
1
,
n
2
,
…
,
n
s
)
=
s
2
∏
i
=
1
s
‍
n
i. For n
1
=
n
2
=
⋯
=
n
s
=
2, we attain the s-dimensional hypercubes Q
s. Therefore, M
1
*
(
Q
s
)
=
s
2
2
s.Remark 15.
Ghorbani and Hosseinzadeh computed the second Zagreb eccentricity index of□
i
=
1
n
G
i in [4]. Here, we can also obtain the explicit formula of M
2
*
(
□
i
=
1
n
G
i
) by induction.
For two connected graphsG
i
(
i
=
1,2
), we note that G
1
(
U
)
⊓
G
2
≅
G
1
□
G
2 if U
=
V
(
G
1
); then by Theorem 5, we have
(18)
ξ
c
(
G
1
□
G
2
)
=
|
G
1
|
ξ
c
(
G
2
)
+
|
G
2
|
ξ
c
(
G
1
)
+
2
|
E
(
G
1
)
|
ζ
(
G
2
)
+
2
|
E
(
G
2
)
|
ζ
(
G
1
)
.
By induction, we can easily prove that(19)
ξ
c
(
□
i
=
1
n
G
i
)
=
∑
i
=
1
n
‍
[
∏
j
=
1
,
j
≠
i
n
‍
|
G
j
|
ξ
c
(
G
i
)
]
+
2
∑
1
≤
s
,
t
≤
n
s
≠
t
[
∏
k
=
1
k
≠
s
,
t
n
|
G
k
|
|
E
(
G
s
)
|
ζ
(
G
t
)
]
.
Therefore, by Corollary9, Lemma 12 and the formula as above, using a similar method of proof in Corollary 13, we can obtain Corollary 16.Corollary 16 (see [4]).
Let□
i
=
1
n
G
i be Cartesian product of n
≥
2 graphs G
i. Then
(20)
M
2
*
(
□
i
=
1
n
G
i
)
=
∑
1
≤
s
,
t
≤
n
s
≠
t
[
∏
k
=
1
k
≠
s
,
t
n
|
G
k
|
(
|
E
(
G
s
)
|
M
1
*
(
G
t
)
+
ξ
c
(
G
s
)
ζ
(
G
t
)
)
]
+
∑
i
=
1
n
‍
[
∏
j
=
1
j
≠
i
n
|
G
j
|
M
2
*
(
G
i
)
+
2
∑
1
≤
s
<
t
≤
n
s
,
t
≠
i

(
∏
k
=
1
k
≠
i
,
s
,
t
n
|
G
j
|
ζ
(
G
s
)
ζ
(
G
t
)
)
|
E
(
G
i
)
|

]
.From Corollaries13 and 16, the following corollary is obvious.Corollary 17.
LetG
n
=
□
i
=
1
n
G
=
G
□
⋯
□
G
︸
n
times be Cartesian product of n
≥
2 connected graphs G. Then
(21)
M
1
*
(
G
n
)
=
n
|
G
|
n
-
2
[
|
G
|
M
1
*
(
G
)
+
(
n
-
1
)
ζ
2
(
G
)
]
,
M
2
*
(
G
n
)
=
2
(
n
2
)
|
G
|
n
-
2
[
|
E
(
G
)
|
M
1
*
(
G
)
+
ξ
c
(
G
)
ζ
(
G
)
]
+
n
|
G
|
n
-
1
M
2
*
(
G
)
+
2
n
(
n
-
1
2
)
|
G
|
n
-
3
ζ
2
(
G
)
.


## 4. Zagreb Eccentricity Indices ofS-Sum Graphs

LetG be a connected graph. The vertices of a Line graph  L
(
G
) are the edges of G. Two edges of G that share a vertex are considered to be adjacent in L
(
G
). A Subdivision graph  S
(
G
) is the graph obtained by inserting an additional vertex in each edge of G. That is, each edge of G is replaced by a path of length two.Definition 18 (see [15]).
For two connected graphsG and H, the S-sum G
+
S
H of G and H is a graph with vertex set (
V
(
G
)
∪
E
(
G
)
)
×
V
(
H
) and vertices (
u
r
,
v
i
) and (
u
s
,
v
k
) are adjacent if and only if [
u
r
=
u
s
∈
V
(
G
) and v
i
v
k
∈
E
(
H
)
] or [
v
i
=
v
k
∈
V
(
H
) and u
r
u
s
∈
E
(
S
(
G
)
)
].Note that if∅
≠
U
=
V
(
G
)
⊂
V
(
S
(
G
)
), then G
+
S
H
≅
S
(
G
)
(
U
)
⊓
H
≅
S
(
G
)
(
V
(
G
)
)
⊓
H. So by Theorem 7, we can compute the Zagreb eccentricity indices of G
+
S
H easily.Lemma 19 (see [15]).
LetG and H be two connected graphs. If U
=
V
(
G
), then we have(a)
|
V
(
S
(
G
)
)
|
=
|
G
|
+
|
E
(
G
)
|, |
E
(
S
(
G
)
)
|
=
2
|
E
(
G
)
|,
(b)
for each vertexv
∈
U, we have d
S
(
G
)
(
v
)
=
d
G
(
v
),
(c)
for each vertexv
∈
V
(
S
(
G
)
)
∖
U, we have d
S
(
G
)
(
v
)
=
2.Lemma 20 (see [15]).
LetG
=
T
n(n
≥
2) be a tree with n vertices. If U
=
V
(
G
), then(a)
for each vertexv
∈
U, we have ε
S
(
G
)
(
U
)
(
v
)
=
2
ε
G
(
v
),
(b)
for each vertexv
∈
V
(
S
(
G
)
)
∖
U, we have ε
S
(
G
)
(
U
)
(
v
)
=
2
ε
L
(
G
)
(
v
)
+
1.Theorem 21.
LetT
n be a tree with order n (n
≥
2) and let H be a connected graph; ∅
≠
U
=
V
(
T
n
)
⊂
V
(
S
(
T
n
)
). Then
(i)(22)
M
1
*
(
T
n
+
S
H
)
=
|
H
|
[
4
M
1
*
(
T
n
)
+
4
M
1
*
(
L
(
T
n
)
)
+
4
ζ
(
L
(
T
n
)
)
+
n
-
1
]
+
(
2
n
-
1
)
M
1
*
(
H
)
+
2
ζ
(
H
)
×
[
2
ζ
(
T
n
)
+
2
ζ
(
L
(
T
n
)
)
+
n
-
1
]
,
(ii)(23)
M
2
*
(
T
n
+
S
H
)
=
4
|
E
(
H
)
|
M
1
*
(
T
n
)
+
2
(
n
-
1
)
M
1
*
(
H
)
+
n
M
2
*
(
H
)
+
2
ξ
c
(
H
)
ζ
(
T
n
)
+
2
|
H
|
[
2
ρ
(
T
n
)
+
ξ
c
(
T
n
)
]
+
2
ζ
(
H
)
[
ξ
c
(
T
n
)
+
2
ζ
(
L
(
T
n
)
)
+
n
-
1
]
,
where ρ
(
T
n
)
=
∑
u
v
∈
E
(
T
n
)
ε
L
(
T
n
)
(
u
v
)
[
ε
T
n
(
u
)
+
ε
T
n
(
v
)
].Proof.
(i) We start to calculateM
1
*
(
T
n
+
S
H
). By Lemmas 19 and 20, we have
(24)
M
1
*
(
S
(
T
n
)
(
U
)
)
=
∑
v
∈
V
(
S
(
T
n
)
)
ε
S
(
T
n
)
(
U
)
2
(
v
)
=
4
∑
v
∈
V
(
T
n
)
ε
T
n
2
(
v
)
+
∑
v
∈
V
(
L
(
T
n
)
)
[
2
ε
L
(
T
n
)
(
v
)
+
1
]
2
=
4
M
1
*
(
T
n
)
+
4
M
1
*
(
L
(
T
n
)
)
+
4
ζ
(
L
(
T
n
)
)
+
n
-
1
,
ζ
(
S
(
T
n
)
(
U
)
)
=
∑
v
∈
V
(
S
(
T
n
)
)
ε
S
(
T
n
)
(
U
)
(
v
)
=
2
∑
v
∈
V
(
T
n
)
ε
T
n
(
v
)
+
2
∑
v
∈
V
(
L
(
T
n
)
)
ε
L
(
T
n
)
(
v
)
+
|
E
(
T
n
)
|
=
2
ζ
(
T
n
)
+
2
ζ
(
L
(
T
n
)
)
+
n
-
1
.
Combing these with (8) in Theorem 7, we can obtain the corresponding result.
(ii) Now, let us computeM
2
*
(
T
n
+
S
H
). By Lemmas 19 and 20, we get
(25)
m
1
*
(
S
(
T
n
)
(
U
)
)
=
∑
v
∈
U
=
V
(
T
n
)
ε
S
(
T
n
)
(
U
)
2
(
v
)
=
4
∑
v
∈
V
(
T
n
)
ε
T
n
2
(
v
)
=
4
M
1
*
(
T
n
)
,
M
2
*
(
S
(
T
n
)
(
U
)
)
=
∑
u
v
∈
E
(
T
n
)
ε
S
(
T
n
)
(
U
)
(
u
v
)
×
[
ε
S
(
T
n
)
(
U
)
(
u
)
+
ε
S
(
T
n
)
(
U
)
(
v
)
]
=
2
∑
u
v
∈
E
(
T
n
)
[
2
ε
L
(
T
n
)
(
u
v
)
+
1
]
×
[
ε
T
n
(
u
)
+
ε
T
n
(
v
)
]
=
4
∑
u
v
∈
E
(
T
n
)
ε
L
(
T
n
)
(
u
v
)
[
ε
T
n
(
u
)
+
ε
T
n
(
v
)
]
+
2
∑
u
v
∈
E
(
T
n
)
[
ε
T
n
(
u
)
+
ε
T
n
(
v
)
]
=
4
ρ
(
T
n
)
+
2
ξ
c
(
T
n
)
,
ξ
c
(
S
(
T
n
)
(
U
)
)
=
∑
u
v
∈
E
(
S
(
T
n
)
)
[
ε
S
(
T
n
)
(
U
)
(
u
)
+
ε
S
(
T
n
)
(
U
)
(
v
)
]
=
∑
v
∈
V
(
S
(
T
n
)
)
d
S
(
T
n
)
(
U
)
(
v
)
ε
S
(
T
n
)
(
U
)
(
v
)
=
2
∑
v
∈
U
=
V
(
T
n
)
d
T
n
(
v
)
ε
T
n
(
v
)
+
2
∑
v
∈
V
(
L
(
T
n
)
)
(
2
ε
L
(
T
n
)
(
v
)
+
1
)
=
2
ξ
c
(
T
n
)
+
4
ζ
(
L
(
T
n
)
)
+
2
(
n
-
1
)
,
ϵ
(
S
(
T
n
)
(
U
)
)
=
∑
v
∈
U
=
V
(
T
n
)
ε
S
(
T
n
)
(
U
)
(
v
)
=
2
∑
v
∈
V
(
T
n
)
ε
T
n
(
v
)
=
2
ζ
(
T
n
)
.
Combing these results with (9) in Theorem 7, we obtain the desired result.Clearly, ifn is even, then ρ
(
P
n
)
=
(
1
/
2
)
n
(
n
-
2
)
+
2
∑
i
=
(
n
/
2
)
n
-
2
‍
i
(
2
i
+
1
). Otherwise, if n is odd, then ρ
(
P
n
)
=
2
∑
i
=
(
(
n
-
1
)
/
2
)
n
-
2
‍
i
(
2
i
+
1
). So, the following lemma holds.Lemma 22.
LetP
n be a path of order n. Then
(26)
ρ
(
P
n
)
=
{
1
12
(
n
-
2
)
(
14
n
2
-
23
n
+
12
)
,
2
∣
n
,
1
12
(
n
-
1
)
(
14
n
2
-
37
n
+
21
)
,
2
∤
n
.Example 23.
SupposeL
n is a linear hexagonal chain with n
≥
2 hexagons (see Figure 1); note that L
n
≅
P
n
+
1
+
S
P
2. Thus, by Lemmas 3 and 22 and Theorem 21, we have
(27)
M
1
*
(
L
n
)
=
2
3
(
14
n
3
+
30
n
2
+
19
n
+
3
)
,
M
2
*
(
L
n
)
=
{
1
3
(
35
n
3
+
57
n
2
+
25
n
+
3
)
,
2
∣
n
,
1
3
(
35
n
3
+
57
n
2
+
28
n
+
6
)
,
2
∤
n
.Figure 1
The linear hexagonal chain withn hexagons L
n.Letn be an integer with n
≥
3 and U
=
V
(
C
n
). Note that ε
S
(
C
n
)
(
U
)
(
v
)
=
n for any vertex v
∈
V
(
S
(
C
n
)
). Then by Theorem 7, we can obtain the following theorem.Theorem 24.
LetC
n  (n
≥
3) be a cycle and let H be an arbitrary connected graph; ∅
≠
U
=
V
(
C
n
)
⊂
V
(
S
(
C
n
)
). Then
(28)
M
1
*
(
C
n
+
S
H
)
=
2
n
(
n
2
|
H
|
+
M
1
*
(
H
)
+
2
n
ζ
(
H
)
)
,
M
2
*
(
C
n
+
S
H
)
=
n
[
(
|
E
(
H
)
|
+
2
|
H
|
)
n
2
+
(
4
ζ
(
H
)
+
ξ
c
(
H
)
)
n
(
|
E
(
H
)
|
+
2
|
H
|
)
n
2
+
2
M
1
*
(
H
)
+
M
2
*
(
H
)
]
.Example 25.
Letn be an integer with n
≥
3 and let Γ be the zigzag polyhex nanotube T
U
H
C
6
[
2
n
,
2
] (see Figure 2); then Γ
≅
C
n
+
S
P
2. By Theorem 24 and Lemma 3, we have M
1
*
(
Γ
)
=
4
n
(
n
+
1
)
2 and M
2
*
(
Γ
)
=
5
n
(
n
+
1
)
2.Figure 2
The zig-zay polyhex nanotubeT
U
H
C
6
[
2
n
,
2
].


## 5. Zagreb Eccentricity Indices of Cluster and Corona Product Graphs

The cluster product, corona product, and join of two graphs are important graph operations defined as below.Definition 26 (see [20]).
The cluster product graphG
{
H
} is obtained by taking one copy of G and |
G
| copies of a rooted graph H and by identifying the root of the ith copy of H with the ith vertex of G, i
=
1,2
,
…
,
|
G
|.Definition 27 (see [20]).
The corona product graphG
⊙
H is obtained by taking one copy of G and |
G
| copies of H and by joining each vertex of the ith copy of H to the ith vertex of G, i
=
1,2
,
…
,
|
G
|.Definition 28 (see [20]).
The join graphG
+
H: V
(
G
+
H
)
=
V
(
G
)
∪
V
(
H
); E
(
G
+
H
)
=
E
(
G
)
∪
E
(
H
)
∪
{
(
u
,
v
)
∣
u
∈
V
(
G
)
,
v
∈
V
(
H
)
}.LetG and H be connected graphs; x is a root-vertex of H. Note that; if ∅
≠
U
=
{
x
}
⊂
V
(
H
), then G
{
H
}
≅
H
(
U
)
⊓
G
≅
H
(
{
x
}
)
⊓
G. We define ω
x
(
H
)
=
∑
u
v
∈
E
(
H
)
‍
d
H
(
u
,
x
)
d
H
(
v
,
x
) and ς
x
(
H
)
=
∑
u
v
∈
E
(
H
)
‍
[
d
H
(
u
,
x
)
+
d
H
(
v
,
x
)
]
=
∑
v
∈
V
(
H
)
‍
d
H
(
v
)
d
H
(
v
,
x
).Theorem 29.
LetG and H be two connected graphs; x is a root-vertex of H. Then
(29)
ξ
c
(
G
{
H
}
)
=
ξ
c
(
G
)
+
2
|
E
(
H
)
|
ζ
(
G
)
+
2
[
|
E
(
G
)
|
+
|
G
|
|
E
(
H
)
|
]
ε
H
(
x
)
+
|
G
|
ς
x
(
H
)
,
(30)
M
1
*
(
G
{
H
}
)
=
|
H
|
M
1
*
(
G
)
+
2
[
|
H
|
ε
H
(
x
)
+
d
(
x
∣
H
)
]
ζ
(
G
)
+
|
G
|
×
[
|
H
|
ε
H
2
(
x
)
+
2
ε
H
(
x
)
d
(
x
∣
H
)
+
d
2
(
x
∣
H
)
]
,
(31)
M
2
*
(
G
{
H
}
)
=
ε
H
(
x
)
ξ
c
(
G
)
+
q
′
M
1
*
(
G
)
+
M
2
*
(
G
)
+
[
ς
x
(
H
)
+
2
|
E
(
H
)
|
ε
H
(
x
)
]
ζ
(
G
)
+
[
|
E
(
G
)
|
+
|
G
|
|
E
(
H
)
|
]
ε
H
2
(
x
)
+
|
G
|
ε
H
(
x
)
ς
x
(
H
)
+
|
G
|
ω
x
(
H
)
.Proof.
Let∅
≠
U
=
{
x
}
⊂
V
(
H
). Then
(32)
ξ
c
(
H
(
U
)
)
=
ξ
c
(
H
(
{
x
}
)
)
=
∑
u
v
∈
E
(
H
)
[
ε
H
(
{
x
}
)
(
u
)
+
ε
H
(
{
x
}
)
(
v
)
]
=
∑
u
v
∈
E
(
H
)
[
d
H
(
u
,
x
)
+
d
H
(
u
,
x
)
+
2
ε
H
(
x
)
]
=
ς
x
(
H
)
+
2
|
E
(
H
)
|
ε
H
(
x
)
.
On the other hand, ϵ
(
H
(
U
)
)
=
ϵ
(
H
(
{
x
}
)
)
=
∑
v
=
x
‍
ε
H
(
{
x
}
)
(
v
)
=
ε
H
(
x
). Note that G
{
H
}
≅
G
⊓
H
(
{
x
}
). Thus, combing these results with Theorem 5, we can obtain (29).
Similarly, we can determine (30) and (31) in terms of Theorem 7, respectively.LetG and H be two simple graphs. If |
G
|
=
n and |
E
(
G
)
|
=
q, then we say that G is an (
n
,
q
)-graph. According to the definitions of the cluster and corona products, if G is an (
n
,
q
)-graph and H is a (
n
′
,
q
′
)-graph, then G
{
H
} is an (
n
n
′
,
q
+
n
q
′
)-graph and G
⊙
H is an (
n
n
′
+
n
,
q
+
n
q
′
+
n
n
′
)-graph.Corollary 30.
LetG be a connected (
n
,
q
)-graph and H is an arbitrary (
n
′
,
q
′
)-graph. Then
(33)
ξ
c
(
G
⊙
H
)
=
ξ
c
(
G
)
+
2
(
q
′
+
n
′
)
ζ
(
G
)
+
2
q
+
4
n
q
′
+
3
n
n
′
,
(34)
M
1
*
(
G
⊙
H
)
=
(
n
′
+
1
)
M
1
*
(
G
)
+
2
(
2
n
′
+
1
)
ζ
(
G
)
+
n
(
4
n
′
+
1
)
,
(35)
M
2
*
(
G
⊙
H
)
=
ξ
c
(
G
)
+
(
n
′
+
q
′
)
M
1
*
(
G
)
+
M
2
*
(
G
)
+
(
3
n
′
+
4
q
′
)
ζ
(
G
)
+
2
n
(
n
′
+
2
q
′
)
+
q
.Proof.
For any(
n
′
,
q
′
)-graph H, let x be the root-vertex of graph H
+
x (the join of graphs H and K
1, x is the unique vertex in K
1). Then H
+
x is a (
n
′
+
1
,
n
′
+
q
′
)-graph. It is easy to see that ε
H
+
x
(
x
)
=
1, ς
x
(
H
+
x
)
=
n
′
+
2
q
′. Note that G
⊙
H
≅
G
{
H
+
x
}; hence ξ
c
(
G
⊙
H
)
=
ξ
c
(
G
{
H
+
x
}
). Equation (33) is obtained by (29). Moreover, we note that d
(
x
∣
H
+
x
)
=
d
2
(
x
∣
H
+
x
)
=
n
′ and ω
x
(
H
+
x
)
=
q
′. Hence, by (30) and (31), using the same method as above, the corresponding equations (34) and (35) are also obtained, respectively.As applications, we present some examples as below, these results can be attained by means of Corollary30, Lemmas 2, and 3.Example 31.
The following equations hold:(36)
ξ
c
(
P
m
⊙
P
n
)
=
{
3
m
2
n
+
5
m
n
-
4
m
,
2
∣
m
,
m
≥
2
;
3
m
2
n
+
5
m
n
-
4
m
-
n
,
2
∤
m
,
m
≥
3
,
ξ
c
(
C
m
⊙
C
n
)
=
m
(
2
(
2
n
+
1
)
⌊
m
2
⌋
+
7
n
+
2
)
,
m
≥
3
,
n
≥
3
,
ξ
c
(
P
m
⊙
C
n
)
=
{
1
2
(
6
m
2
n
+
3
m
2
+
10
m
n
-
2
m
)
,

2
∣
m
,
m
≥
2
,
n
≥
3
;

1
2
(
6
m
2
n
+
3
m
2
+
10
m
n
-
2
m
-
2
n
-
1
)
,

2
∤
m
,
m
≥
3
,
n
≥
3
,

ξ
c
(
C
m
⊙
P
n
)
=
m
(
4
n
⌊
m
2
⌋
+
7
n
-
2
)
,
m
≥
3
.Example 32.
LetH be an arbitrary graph with n vertices. Then
(37)
M
1
*
(
P
m
⊙
H
)
=
{
1
12
m
(
7
m
2
n
+
7
m
2
+
27
m
n
+
9
m
+
26
n
+
2
)
,

2
∣
m
,
m
≥
2
;

1
12
(
7
m
3
n
+
7
m
3
+
27
m
2
n
+
9
m
2
-
m
-
9
n
-
3
)
,

2
∤
m
,
m
≥
3
,

(38)
M
1
*
(
C
m
⊙
H
)
=
m
(
(
n
+
1
)
⌊
m
2
⌋
2
+
2
(
2
n
+
1
)
⌊
m
2
⌋
+
4
n
+
1
)
,
m
≥
3
.Example 33.
The following equations hold:(39)
M
2
*
(
P
m
⊙
P
n
)
=
{
1
12
(
14
m
3
n
+
45
m
2
n
-
30
m
2
+
34
m
n
-
30
m
+
12
14
m
3
n
+
45
m
2
n
-
30
m
2
)
,
2
∣
m
,
m
≥
2
;
1
12
(
14
m
3
n
+
45
m
2
n
-
30
m
2
+
28
m
n
-
30
m
-
15
n
+
12
14
m
3
n
+
45
m
2
n
-
30
m
2
)
,
2
∤
m
,
m
≥
3
,
M
2
*
(
C
m
⊙
C
n
)
=
m
(
(
2
n
+
1
)
⌊
m
2
⌋
2
+
(
7
n
+
2
)
⌊
m
2
⌋
+
6
n
+
1
)
,
m
≥
3
,
n
≥
3
,
M
2
*
(
P
m
⊙
C
n
)
=
{
1
12
(
14
m
3
n
+
7
m
3
+
45
m
2
n
-
3
m
2
+
34
m
n
-
4
m
+
12
14
m
3
n
+
45
m
2
n
-
30
m
2
)
,
2
∣
m
,
m
≥
2
,
n
≥
3
;
1
12
(
14
m
3
n
+
7
m
3
+
45
m
2
n
-
3
m
2
+
28
m
n
-
7
m
-
15
n
+
3
14
m
3
n
+
45
m
2
n
-
30
m
2
)
,
2
∤
m
,
m
≥
3
,
n
≥
3
,
M
2
*
(
C
m
⊙
P
n
)
=
m
(
2
n
⌊
m
2
⌋
2
+
(
7
n
-
2
)
⌊
m
2
⌋
+
6
n
-
3
)
,
m
≥
3
.



---

*Source: 241712-2014-02-10.xml*