Faculty of Sciences and Mathematics, University of Niˇs, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Filomat 26:1 (2012), 93–100 DOI: 10.2298/FIL1201093G
A new version of Zagreb indices
Modjtaba Ghorbani
∗
, Mohammad A. Hosseinzadeh
Abstract
The Zagreb indices have been introduced by Gutman and Trinajsti´c as
M
1
(G) =
X
v∈V (G)
(d
G
(v))
2
and M
2
(G) =
X
uv∈E(G)
d
G
(u)d
G
(v), where d
G
(u) de-
notes the degree of vertex u. We now define a new version of Zagreb indices
as M
∗
1
(G) =
X
uv∈E(G)
[ε
G
(u) + ε
G
(v)] and M
∗
2
(G) =
X
uv∈E(G)
ε
G
(u)ε
G
(v), where
ε
G
(u) is the largest distance between u and any other vertex v of G. The goal
of this paper is to further the study of these new topological index.
1 Introduction
A graph is a collection of points and lines connecting a subset of them. The points
and lines of a graph are also called vertices and edges of the graph, respectively.
The vertex and edge sets of a graph G are denoted by V (G) and E(G), respectively.
A molecular graph is a simple graph such that its vertices correspond to the atoms
and the edges to the bonds. Note that hydrogen atoms are often omitted. Chemical
graph theory is a branch of mathematical chemistry which has an important effect
on the development of the chemical sciences.
By IUPAC terminology, a topological index is a numerical value associated with
chemical constitution purporting for correlation of chemical structure with various
physical properties, chemical reactivity or biological activity. In an exact phrase, if
Graph denotes the class of all finite graphs then a topological index is a function
T op from Graph into real numbers with this property that T op(G) = T op(H), if
G and H are isomorphic. Obviously, the number of vertices and the number of
edges are topological index. The Wiener index [13] is the first reported distance
based topological index defined as half sum of the distances between all the pairs
of vertices in a molecular graph.
If x, y ∈ V (G) then the distance d
G
(x, y) between x and y is defined as the
length of any shortest path in G connecting x and y. For a vertex u of V (G) its
∗
Corresponding Author: mghorbani@srttu.edu
2010 Mathematics Subject Classifications 05C40, 05C90.
Key words and Phrases Zagreb indices, composite graph, product graph.
Received: September 10, 2010
Communicated by Dragan S. Djordjevi´c
94 M. Ghorbani, M. A. Hosseinzadeh
eccentricity ε
G
(u) is the largest distance between u and any other vertex v of G,
ε
G
(u) = max
v ∈V (G)
d
G
(u, v). The maximum eccentricity over all vertices of G is
called the diameter of G and denoted by D(G). The eccentric connectivity index
ξ(G) of a graph G is defined as
ξ(G) =
X
u∈V (G)
d
G
(u)ε
G
(u),
where d
G
(u) denotes the degree of vertex u in G, i. e., the number of its neighbors
in G. When the vertex degrees are not taken into account, we obtain the total
eccentricity of the graph G, ζ(G) =
P
u∈V (G)
ε
G
(u). For k-regular graphs those two
quantities are related as ξ(G) = kζ(G). We refer the reader to [2, 4, 6, 9, 15] for
explicit formulas for the eccentric connectivity index of various families of graphs.
A vertex u ∈ V (G) is well-connected if ε
G
(u) = 1, i.e., if it is adjacent to all other
vertices in G.
The Zagreb indices have been introduced more than thirty years ago by Gutman
and Trinajsti´c [8, 7]. They are defined as:
M
1
(G) =
P
v ∈V (G)
(d
G
(v))
2
and M
2
(G) =
P
uv ∈E(G)
d
G
(u)d
G
(u).
Now we define a new version of Zagreb indices as follows:
M
∗
1
(G) =
X
uv ∈E(G)
[ε
G
(u) + ε
G
(v)],
M
∗∗
1
(G) =
X
v ∈V (G)
(ε
G
(v))
2
,
M
∗
2
(G) =
X
uv ∈E(G)
ε
G
(u)ε
G
(v).
Here, our notation is standard and mainly taken from standard books of graph
theory such as, e.g., [14]. All graphs considered in this paper are simple and con-
nected. The aim of this paper is to compute these new topological indices for some
graph operations. To do this, we first consider the following examples:
Example 1. Let K
n
be the complete graph on n vertices. Then for every v ∈
V (K
n
), ε
G
(v) = 1. This implies that ζ(K
n
) = n, M
∗
1
(K
n
) = n(n − 1), M
∗
2
(K
n
) =
n(n − 1)/2 and M
∗∗
1
(K
n
) = n.
Example 2. Let C
n
denote the cycle of length n. It is easy to see that for
every v ∈ V (C
n
), ε
G
(v) = bn/2c. Hence, ζ(C
n
) = nb
n
2
c, M
∗
1
(C
n
) = 2nbn/2c and
M
∗∗
1
(C
n
) = M
∗
2
(C
n
) = nbn/2c
2
.
Example 3. Let S
n
= K
1,n
be the star graph with n + 1 vertices. The central
vertex has degree n and eccentricity 1, while the remaining n vertices have degree
A new version of Zagreb indices 95
1 and eccentricity 2. Hence, ζ(S
n
) = 2n + 1, M
∗
1
(S
n
) = 3n, M
∗
2
(S
n
) = 2n and
M
∗∗
1
(S
n
) = 4n + 1.
Example 4. A wheel W
n
is a graph of order n + 1 which contains a cycle on
n vertices and a central vertex connected to each vertex of the cycle. Again, the
central vertex has degree n and eccentricity 1, while the peripheral vertices have
degree 3 and eccentricity 2. So, ζ(W
n
) = 2n + 1, M
∗
1
(W
n
) = 7n, M
∗
2
(W
n
) = 6n and
M
∗∗
1
(W
n
) = 4n + 1.
Example 5. Let P
n
be the path on n ≥ 3 vertices. Then
ζ(P
n
) =
n(3n − 2)/4 2|n
(n − 1)(3n + 1)/4 2 6 |n
,
M
∗
1
(P
n
) =
(3n
2
− 6n + 4)/2 2|n
3(n − 1)
2
/2 2 6 |n
,
M
∗
2
(P
n
) =
n(n − 2)(7n − 10)/12 + n
2
/4 2|n
(n − 1)(7n
2
− 14n + 3)/12 2 6 |n
,
M
∗∗
1
(P
n
) =
n(n − 1)(7n − 2)/12 2|n
(n − 1)(7n
2
− 2n − 3)/12 2 6 |n
.
2 Main Results
In this section we define some graph operations [10] and then we compute the Zagreb
indices for them.
Cartesian product
The Cartesian product of two graphs G
1
and G
2
is denoted by G
1
G
2
has
the vertex set V (G
1
) × V (G
2
) and, two vertices u = (u
1
, u
2
) and v = (v
1
, v
2
)
are connected by an edge if and only if either ([u
1
= v
1
and u
2
v
2
∈ E(G
2
)]) or
([u
2
= v
2
and u
1
v
1
∈ E(G
1
)]). In other word, |E(G
1
G
2
)| = |E(G
1
)||V (G
2
)| +
|E(G
2
)||V (G
1
)|. The degree of a vertex (u
1
, u
2
) of G
1
G
2
is as follows:
d
G
1
G
2
(u
1
, u
2
) = d
G
1
(u
1
) + d
G
2
(u
2
).
Lemma 6. ε
G
1
G
2
(u
1
, u
2
) = ε
G
1
(u
1
) + ε
G
2
(u
2
).
96 M. Ghorbani, M. A. Hosseinzadeh
Proof. It is clear that the eccentricity of a vertex (u
1
, u
2
) ∈ V (G
1
G
2
) cannot
exceed the sum of the eccentricities of its projections u
1
and u
2
. On the other
hand, this upper bound is attained for (w
1
, w
2
), where w
i
is the vertex on which
ε(u
i
) is attained, for i = 1, 2. This proves the claim.
The Cartesian product of more than two graphs is denoted by
Q
s
i=1
G
i
, in which
Q
s
i=1
G
i
= G
1
. . . G
s
= (G
1
. . . G
s−1
)G
s
. If G
1
= G
2
= ... = G
s
= G, we
have the s-th Cartesian power of G and denote it by G
s
.
Lemma 7.ε
k
i=1
G
i
((u
1
, · · · , u
k
)) =
k
X
i=1
ε
G
i
(u
i
).
Theorem 8.
M
∗
2
(
n
k=1
G
i
) =
n
X
k=1
M
∗
1
(G
k
)
n
X
i=1,i6=k
n
Y
j=1,j6=i,k
|V (G
j
)|ζ(G
i
)
+
n
X
k=1
|E(G
k
)|
n
X
i=1,i6=k
n
Y
j=1,j6=i,k
|V (G
j
)|M
∗∗
1
(G
i
)
+
n
X
k=1
M
∗
2
(G
k
)
n
Y
i=1,i6=k
|V (G
i
)|
+ 2
n
X
k=1
|E(G
k
)|
i,j6=k
X
1≤i<j≤n
n
Y
r=1,r6=i,j,k
|V (G
r
)|ζ(G
i
)ζ(G
j
).
Proof. Let a = (a
1
, · · · , a
k
) and b = (b
1
, · · · , b
k
). Then we have
M
∗
2
(
n
k=1
G
i
) =
X
ab∈E(
n
k=1
G
i
)
ε
n
k=1
G
i
(a)ε
n
k=1
G
i
(b)
=
n
X
k=1
X
a
1
∈V (G
1
)
· · ·
X
a
k
b
k
∈E(G
k
)
· · ·
X
a
n
∈V (G
n
)
ε
G
k
(a
k
)
+ ε
G
k
(b
k
)
n
X
i=1,i6=k
ε
G
i
(a
i
)
+
n
X
i=1,i6=k
(ε
G
i
(a
i
))
2
+
n
X
k=1
X
a
1
∈V (G
1
)
· · ·
X
a
k
b
k
∈E(G
k
)
· · ·
X
a
n
∈V (G
n
)
ε
G
k
(a
k
)ε
G
k
(b
k
)
+ 2
i,j6=k
X
1≤i<j≤n
ε
G
i
(a
i
)ε
G
j
(b
j
)
.
A new version of Zagreb indices 97
Corollary 9. Let G and H be graphs. Then
M
∗
2
(GH) = M
∗
1
(G)ζ(H) + M
∗
1
(H)ζ(G) + |V (H)|M
∗∗
1
(G) + |V (G)|M
∗∗
1
(H)
+ |V (H)|M
∗
2
(G) + |V (G)|M
∗∗
1
(H).
Example 10. A Hamming graph H
n
1
,n
2
,··· ,n
s
is defined as H
n
1
,n
2
,··· ,n
s
=
s
i=1
K
n
i
.
So, M
∗
2
(H
n
1
,n
2
,··· ,n
s
) =
s
X
k=1
s
2
n
k
2
s
Y
i=1,i6=k
n
i
. For n
1
= n
2
· · · = n
s
= 2, we achieve
the s-dimensional hypercubes Q
s
and so, M
∗
2
(Q
s
) = s
3
2
s−1
.
Example 11. Nanotubes and nanotori covered by C
4
are arisen as Cartesian
product of a path and a cycle, two cycles, respectively. By Combining examples 2
and 5 with Corollary 9 we obtain the following explicit formulas for nanotubes and
nanotori. We denote R = P
n
C
m
and S = C
k
C
m
and assume n ≥ 3. Then
M
∗
2
(R) =
(2n − 1)mbm/2c
2
+ (3n
2
− 4n + 2)mbm/2c
+nm(7n
2
− 15n + 11)/6 2|n
(2n − 1)mbm/2c
2
+ (n − 1)(3n − 1)mbm/2c
+nm(n − 1)(7n − 8)/6 2 6 |n
,
M
∗
2
(S) = 2km
bm/2c
2
+ bk/2c
2
+ 2bm/2cbk/2c
.
Disjunction and Symmetric Difference
The disjunction G
1
∨ G
2
of two graphs G
1
and G
2
is the graph with vertex set
V (G
1
) × V (G
2
) in which (u
1
, u
2
) is adjacent to (v
1
, v
2
) whenever u
1
is adjacent to
v
1
in G
1
or u
2
is adjacent to v
2
in G
2
. So,
|E(G ∨ H)| = |E(G)||V (H)|
2
+ |E(H)||V (G)|
2
− 2|E(G)||E(H)|.
The symmetric difference G
1
⊕G
2
of two graphs G
1
and G
2
is the graph with vertex
set V (G
1
) × V (G
2
) in which (u
1
, u
2
) is adjacent to (v
1
, v
2
) whenever u
1
is adjacent
to v
1
in G
1
or u
2
is adjacent to v
2
in G
2
, but not both. From definition one can see
that
|E(G
1
⊕ G
2
)| = |E(G
1
)||V (G
2
)|
2
+ |E(G
2
)||V (G
1
)|
2
− 4|E(G
1
)||E(G
2
)|.
The distance between any two vertices of a disjunction or a symmetric difference
cannot exceed 2. If none of the components contains well-connected vertices, the
eccentricity of all vertices is constant and equal to 2.
Lemma 12. Let G
1
and G
2
be two graphs without well-connected vertices. Then
ε
G
1
⊕G
2
((u
1
, u
2
)) = ε
G
1
∨G
2
((u
1
, u
2
)) = 2.
Theorem 13. Let G and H be two graphs without well-connected vertices. Then
M
∗
2
(G ∨ H) = 4
|E(G)||V (H)|
2
+ |E(H)||V (G)|
2
− 2|E(G)||E(H)|
,