DOI: 10.1515/auom-2016-0008
An. Şt. Univ. Ovidius Constanţa
Vol. 24(1),2016, 153–176
On the first Zagreb index and multiplicative
Zagreb coindices of graphs
Kinkar Ch. Das, Nihat Akgunes, Muge Togan, Aysun Yurttas,
I. Naci Cangul, A. Sinan Cevik
Abstract
For a (molecular) graph G with vertex set V (G) and edge
P set E(G),
the first Zagreb index of G is defined as M1 (G) =
dG (vi )2 ,
vi ∈V (G)
where dG (vi ) is the degree of vertex vi in G. Recently Xu et al. introY
Y
duced two graphical invariants
(G) =
(dG (vi )+dG (vj )) and
1
Y
2
(G) =
Y
vi vj ∈E(G)
/
dG (vi ) dG (vj ) named as first multiplicative Zagreb
vi vj ∈E(G)
/
coindex and second multiplicative Zagreb coindex, respectively. The
Narumi-Katayama index of a graph G, denoted by N K(G), is equal
to the product of the degrees of the vertices of G, that is, N K(G) =
n
Y
dG (vi ) . The irregularity index t(G) of G is defined as the numi=1
ber of distinct terms in the degree sequence of G. In this paper, we
give some lower and upper bounds on the first Zagreb index M1 (G) of
graphs and trees in terms of number of vertices, irregularity index, maximum degree, and characterize the extremal graphs. Moreover, we obtain
some lower and upper bounds on the (first and second) multiplicative
Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and NarumiKatayama index, and (first and second) multiplicative Zagreb index and
coindices of graphs.
Key Words: First Zagreb index, First and Second multiplicative Zagreb coindex,
Narumi-Katayama index.
2010 Mathematics Subject Classification: Primary 05C07; Secondary 05C12.
Received: 3.05.2014
Accepted: 2.10.2014.
153
On the first Zagreb index and multiplicative Zagreb coindices of graphs
1
154
Introduction
We only consider finite, undirected and simple graphs throughout this paper.
Let G be a graph with vertex set V (G) (|V (G)| = n) and edge set E(G)
(|E(G)| = m). For a graph G, we let dG (vi ) be the degree of a vertex vi in G.
The maximum vertex degree in G is denoted by ∆ and the minimum vertex
degree by δ . For each vi ∈ V (G) , the set of neighbors of the vertex vi is
denoted by NG (vi ) . The irregularity index t(G) of G is defined as the number
of distinct terms in the degree sequence of G [1]. Denote by p, the number
of pendent vertices in G. For terminology and notation not defined here, the
reader is referred to Bondy and Murty [2].
Among the oldest and most studied topological indices, there are two classical vertex-degree based topological indices–the first Zagreb index and second
Zagreb index. These two indices first appeared in [14], and were elaborated
in [15]. Later they were used in the structure-property model (see [20]). The
first Zagreb index M1 (G) and the second Zagreb index M2 (G) of a graph G
are defined, respectively, as
X
M1 (G) =
dG (vi )2 ,
vi ∈V (G)
and
M2 (G) =
X
dG (vi ) dG (vj ).
vi vj ∈E(G)
During the past decades, numerous results concerning Zagreb indices have
been put forward, see [3, 5, 7, 9, 13] and the references cited therein.
The Narumi-Katayama index of a graph G, denoted by N K(G), is equal
to the product of the degrees of the vertices of G, that is,
N K(G) =
n
Y
dG (vi ) .
i=1
Recently, the first and second multiplicative Zagreb indices [11, 18, 19] are
defined as follows:
n
Y
Y
(G) =
dG (vi )2
1
i=1
and
Y
2
(G) =
Y
vi vj ∈E(G)
dG (vi ) dG (vj ) =
n
Y
dG (vi )dG (vi ) .
i=1
The mathematical properties of the first and second multiplicative Zagreb
indices have been given in [6, 11].
On the first Zagreb index and multiplicative Zagreb coindices of graphs
155
Xu et al. recently defined the multiplicative Zagreb coindices as follows
[21]:
Y
1
(G) =
Y
(dG (vi )+dG (vj )) and
Y
2
vi vj ∈E(G)
/
(G) =
Y
dG (vi ) dG (vj ) .
vi vj ∈E(G)
/
In the same reference, it has been also given the mathematical properties of
the first and second multiplicative Zagreb coindices.
The paper is organized as follows. In Section 2, we give some lower and
upper bounds on the Zagreb indices of graphs and trees, and characterize
the extremal graphs. In Section 3, we obtain some lower and upper bounds
on multiplicative Zagreb coindices of graphs and characterize the extremal
graphs. In Section 4, we present some relations between first Zagreb index
and Narumi-Katayama index, and (first and second) multiplicative Zagreb
index and coindices of graphs.
2
Bounds on the first Zagreb index of trees and graphs
In this section we give some lower and upper bounds on the first Zagreb index
of trees and graphs in terms of n, t and ∆. For this we need the following
result:
Lemma 1. Let T be a tree of order n with irregularity index t. Then the
number of pendent vertices in T is
p≥
t−1
X
i=1
(dG (vi ) − 2) + 2 =
t−1
X
dG (vi ) − 2t + 4,
(1)
i=1
where dG (v1 ) > dG (v2 ) > · · · > dG (vt−1 ) > dG (vt ) = 1 and dG (vi ) is the
degree of the vertex vi in T . Moreover, the equality holds in (1) if and only if
n = p + t − 1.
Proof. For any graph G, it is well known that
n
X
dG (vi ) = 2m, where m is
i=1
the number of edges in G .
Since T is a tree, we have m = n − 1. Without loss of generality, we can
assume that dT (v1 ) ≥ dT (v2 ) ≥ · · · ≥ dT (vn ) . Let p be the number of pendent
vertices in T . For i = n − p + 1, n − p + 2, . . . , n, we then have dT (vi ) = 1.
On the first Zagreb index and multiplicative Zagreb coindices of graphs
156
Using these results, we get
n−p
X
i=1
⇐⇒
p=
n
X
dT (vi ) +
dT (vi ) = 2(n − 1) ⇐⇒ 2n − p − 2 =
i=n−p+1
n−p
X
n−p
X
dT (vi )
i=1
(dT (vi ) − 2) + 2
i=1
⇐⇒
p≥
t−1
X
(dT (vi ) − 2) + 2 as n − p ≥ t − 1 and dT (vi ) ≥ 2 ,
(2)
i=1
where i = 1, 2, . . . , n − p. Moreover, the equality holds in (2) if and only if
n = p + t − 1. Hence the equality holds in (1) if and only if n = p + t − 1.
Let Γ1 be the class of trees T = (V, E) such that T is a tree of order n,
irregularity index t, maximum degree ∆ and
∆ = t,
dT (vi ) = 1, i = t, t + 1, . . . , n.
s
s
s
s
s
s
s
s
Figure 1:
Example 2. For a tree T1 as in Figure 1, it is clear that n = 8, ∆ = t = 4
and the number of pendent vertices are 5 (= n − t + 1). Moreover, the degree
sequence of tree T1 is (4, 3, 2, 1, 1, 1, 1, 1). Hence T1 ∈ Γ.
Now we are ready to give an upper bound on the first Zagreb index of trees
in terms of n, t and ∆.
Theorem 3. Let T be a tree of order n with irregularity index t and maximum
degree ∆. Then
t(t − 3)
M1 (T ) ≤
n−3−
∆2 − (t − 1)(t − 2)∆
2
1
+ (t3 − 3t2 + 2t + 6)
(3)
3
with equality if and only if G ∈ Γ1 .
On the first Zagreb index and multiplicative Zagreb coindices of graphs
157
Proof. Since the irregularity index of T is t, let us consider a set S(T ) =
{v1 , v2 , . . . , vt−1 } such that ∆ = dG (v1 ) > dG (v2 ) > . . . > dG (vt−1 ) > 1,
where dG (vi ) is the degree of the vertex vi in T . Since dG (vi ) ≤ ∆, for all
vi ∈ V (T ), we have
X
dG (vi )2 =
t−1
X
dG (vi )2
≤
i=1
vi ∈S(T )
t−1
X
(∆ − i + 1)2
i=1
=
t−1
X
(∆ + 1)2 + i2 − 2i(∆ + 1)
i=1
=
1
(t − 1)t(2t − 1)
6
(t − 1)(∆ + 1)2 +
−(∆ + 1)t(t − 1) .
(4)
Moreover, we have
X
dG (vi ) =
t−1
X
dG (vi ) ≥ 2 + 3 + · · · + t =
i=1
vi ∈S(T )
t(t + 1)
− 1.
2
(5)
Let p be the number of pendent vertices in T . Now,
M1 (T )
=
n
X
dG (vi )2
i=1
X
=
dG (vi )2 +
vi ∈V (T ), dG (vi )=1
X
+
X
dG (vi )2
vi ∈S(T )
dG (vi )2
vi ∈V (T )\S(T ), dG (vi )>1
≤ p+
t−1
X
dG (vi )2 + (n − p − t + 1)∆2 since dG (vi ) ≤ ∆
i=1
≤ (n − t + 1)∆2 − p(∆2 − 1) + (t − 1)(∆ + 1)2
1
+ (t − 1)t(2t − 1) − (∆ + 1)t(t − 1) by (4) .
6
(6)
On the first Zagreb index and multiplicative Zagreb coindices of graphs
158
By Lemma 1, the next step of last inequality is
!
t−1
X
2
≤ n∆ −
dG (vi ) − 2t + 4 (∆2 − 1) − (t2 − 3t + 2)∆
i=1
≤
1
3
13
+ t3 − t2 + t − 1
3
6
2
t(t − 3)
2
n∆ −
+ 3 (∆2 − 1) − (t2 − 3t + 2)∆ +
2
1 3 3 2 13
t − t + t − 1 by (5),
3
2
6
(7)
which gives the required result in (3). First part of the proof is completed.
Now suppose that the equality holds in (3). Then all the inequalities in
the above must be equalities. From the equality in (7), we get
n=p+t−1
(by Lemma 1).
From the equality in (4), we get
dG (vi ) = ∆ − i + 1, i = 1, 2, . . . , t − 1.
From the equality in (5), we get
dG (vi ) = t − i + 1, i = 1, 2, . . . , t − 1
From the above results, we finally obtain ∆ = t and n = p + t − 1. Thus
we have
2n − 4 = t(t − 1), ∆ = t and dG (vi ) = 1 for i = t, t + 1, . . . , n.
Hence T ∈ Γ1 .
Conversely, let T ∈ Γ1 . Then we have ∆ = t, p = n − t + 1 and n =
1
t(t
− 1) + 2. We have
2
M1 (T ) = t2 + (t − 1)2 + · · · + 22 + 12 + (n − t) =
and
t(t − 3)
(n − 3) −
∆2
2
−
(t − 1)(t − 2)∆ +
1 3
(t − 3t2 + 2t + 6) =
3
1 3
4
t + t2 − t + 2 .
3
3
This completes the theorem.
1 3
4
t + t2 − t + 2
3
3
On the first Zagreb index and multiplicative Zagreb coindices of graphs
159
Let Γ2 be the class of graphs H1 = (V, E) such that H1 is a graph of order
n, irregularity index t, maximum degree ∆ and
∆ = t, dG (vi ) = 1, i = t + 1, t + 2, . . . , n.
Let Γ3 be the class of graphs H2 = (V, E) such that H2 is a graph of order
n, irregularity index t, maximum degree ∆ and
∆ − i + 1 ; i = 1, 2, . . . , t
dG (vi ) =
.
∆
; i = t + 1, t + 2, . . . , n
r
r
r
s
r
r
r
r
r
r
r
G2
G1
Figure 2:
Example 4. Two graphs G1 and G2 are depicted in Figure 2. For G1 , n = 6,
∆ = t = 4 and the number of pendent vertices are 3 (= n − t + 1). Moreover,
the degree sequence of G1 is (4, 3, 2, 1, 1, 1). Hence G1 ∈ Γ2 . For G2 , n = 5,
∆ = t = 3 and the number of maximum degree vertices are 3 (= n − t + 1).
Moreover, the degree sequence of G2 is (3, 3, 3, 2, 1). Hence G2 ∈ Γ3 .
Now we give some lower and upper bounds on the first Zagreb index M1 (G)
of graphs G in terms of n, t and ∆.
Theorem 5. Let G be a graph of order n with irregularity index t and maximum degree ∆. Then

M1 (G) ≥ 61 t(t + 1)(2t + 1) + n − t



and
(8)
1
2
M1 (G) ≤ t(∆ + 1) + 6 t(t + 1)(2t + 1) 

2 
−(∆ + 1)t(t + 1) + (n − t)∆
with the lower and upper bounds in (8) become equality if and only if G ∈ Γ2
and G ∈ Γ3 , respectively.
Proof. Since the irregularity index of G is t, let us consider a set S(G) =
{v1 , v2 , . . . , vt−1 } such that ∆ = dG (v1 ) > dG (v2 ) > . . . > dG (vt−1 ) > 1,
On the first Zagreb index and multiplicative Zagreb coindices of graphs
160
where dG (vi ) is the degree of the vertex vi in G. Since δ ≤ dG (vi ) ≤ ∆, for all
vi ∈ V (G), we have
M1 (G)
=
n
X
dG (vi )2
i=1
=
X
vi ∈S(G)
≥
t
X
X
dG (vi )2 +
dG (vi )2
vi ∈V (G)\S(G)
j 2 + (n − t) 1 =
j=1
1
t(t + 1)(2t + 1) + n − t
6
and
M1 (G)
=
X
dG (vi )2 +
vi ∈S(G)
≤
X
dG (vi )2
vi ∈V (G)\S(G)
t
X
(∆ − j + 1)2 + (n − t)∆2
j=1
=
t(∆ + 1)2 +
1
t(t + 1)(2t + 1) − (∆ + 1)t(t + 1) + (n − t)∆2 .
6
First part of the proof is done.
Now suppose that there exits an equality for the lower bound in (8). Then
we must have
∆ = t, dG (vi ) = t−i+1, i = 1, 2, . . . , t−1 and dG (vi ) = 1, i = t, t+1, . . . , n.
Hence T ∈ Γ2 . On the other hand, if we suppose the existence of the equality
for the upper bound in (8), then we must have
dG (vi ) = ∆ − i + 1, i = 1, 2, . . . , t and dG (vi ) = ∆, i = t + 1, t + 2, . . . , n
which implies T ∈ Γ3 .
Conversely, by taking G ∈ Γ2 and G ∈ Γ3 respectively, one can easily see
that the truthness of the both equalities in (8).
3
Bounds on the Multiplicative Zagreb coindices of graphs
Let Γ4 be the class of graphs H4 = (V, E) such that H4 is a connected graph
of maximum degree ∆, number of pendent vertices p (p > 0) and
dH4 (v1 ) = dH4 (v2 ) = · · · = dH4 (vn−p ) = ∆
161
On the first Zagreb index and multiplicative Zagreb coindices of graphs
and
dH4 (vn−p+1 ) = dH4 (vn−p+2 ) = · · · = dH4 (vn ) = 1.
r
s
r
s
s
r
r
r
r
G3
r
s
s
r
G4
r
Figure 3:
Example 6. Let G3 and G4 be the graphs depicted in Figure 3. For G3 , n = 8,
∆ = 4, t = 2 and the number of pendent vertices are p = 4. Moreover, the
degree sequence of graph G3 is (4, 4, 4, 4, 1, 1, 1, 1). Hence G3 ∈ Γ4 . For G4 ,
n = 6, ∆ = 3, t = 2 and the number of pendent vertices are p = 2. Moreover,
the degree sequence of G4 is (3, 3, 3, 3, 1, 1). Hence G4 ∈ Γ4 .
Now we give some lower and upper bounds on the first multiplicative Zagreb coindex in terms of n, m, p, ∆ and non-pendent minimum degree δ.
Theorem 7. Let G be a connected graph of order n with m edges, number of
pendent vertices p, maximum degree ∆ and non-pendent minimum degree δ.
− m − p(p−1)
− p(n − p − 1). We
Also, for simplicity, let A denotes n(n−1)
2
2
then have
p(p − 1)
p(n−p−1)
2
(i)
(δ + 1)
(2δ)A ,
1 (G) ≥ 2
p(p − 1)
Q
p(n−p−1)
2
(ii)
(G)
≤
2
(∆ + 1)
(2∆)A .
1
Q
(9)
(10)
The equality holds in both (9) and (10) if and only if G is isomorphic to a
regular graph or G ∈ Γ4 .
Proof. The number of vertex pairs (vi , vj ) such that vi vj ∈
/ E(G) in G is
n(n − 1)
− m. Since G is connected and the number of pendent vertices in G
2
p(p − 1)
is p, then
is the number of vertex pairs (vi , vj ) such that vi vj ∈
/ E(G)
2
with dG (vi ) = dG (vj ) = 1, p(n − p − 1) is the number of vertex pairs (vi , vj )
n(n − 1)
such that vi vj ∈
/ E(G) with dG (vi ) = 1 or dG (vj ) = 1, and
−
2
p(p − 1)
m−
− p(n − p − 1) is the number of vertex pairs (vi , vj ) such that
2
On the first Zagreb index and multiplicative Zagreb coindices of graphs
162
vi vj ∈
/ E(G) with dG (vi ) > 1 and dG (vj ) > 1. From the above, we get
Y
Y
(G) =
(dG (vi ) + dG (vj ))
1
which equals to
Y
vi vj ∈E(G)
/
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
×
Y
Y
(dG (vi ) + dG (vj ))
(dG (vi ) + dG (vj ))
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
(dG (vi ) + dG (vj ))
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
≥
2
p(p − 1)
p(n−p−1)
2
(δ + 1)
(2δ)A ,
where A is defined as in the statement of the theorem. Moreover, the above
equality holds if and only if G is isomorphic to regular graph (p = 0) or G ∈ Γ4
(p > 0).
Q
Similarly, 1 (G) equals to the
Y
Y
(dG (vi ) + dG (vj ))
(dG (vi ) + dG (vj ))
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
×
Y
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
(dG (vi ) + dG (vj ))
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
≤
p(p − 1)
p(n−p−1)
A
2
(∆ + 1)
(2∆) ,
2
where A is defined as in the statement of the theorem. We note that the last
inequality becomes equality if and only if G is isomorphic to regular graph
(p = 0) or G ∈ Γ4 (p > 0).
Hence the result.
The following inequality was obtained by Furuichi [12].
Lemma 8. [12] For a1 , a2 , . . . , an ≥ 0 and p1 , p2 , . . . , pn ≥ 0 satisfying
n
X
i=1
pi = 1,
On the first Zagreb index and multiplicative Zagreb coindices of graphs
163
there exists
n
X
pi ai −
i=1
n
Y
api i
≥ nλ
i=1
n
n
Y
1X
1/n
ai −
ai
n i=1
i=1
!
,
(11)
where λ = min{p1 , p2 , . . . , pn }. Moreover, equality in (11) holds if and only if
a1 = a2 = · · · = an .
Q
Now we obtain an upper bound on 2 (G) of graph G in terms of n, m, p,
the first Zagreb index M1 (G) and the Narumi-Katayama index N K(G) .
Theorem 9. Let G be a connected graph of order n, m edges, p pendent
vertices and maximum degree ∆. Then
"
Y
(n − p − 1)(2m − p) + p − M1 (G)
(G) ≤ (N K(G))p
2
(n − p)(n − p − 1) − 2m + p
−
(n − p)(n − ∆ − p − 1)
(n − p)(n − p − 1) − 2m + p
×
2m − p
− (N K(G))1/(n−p)
n−p
#(n−p)(n−p−1)−2m+p
. (12)
Additionally, the equality holds in (12) if and only if G ∈ Γ4 or G is isomorphic
to a regular graph.
Proof. Since p is the number of pendent vertices in G, we need to consider the
proof in two cases as in the following.
Case (i) : The situation p > 0:
We can assume that
dG (vn−p+1 ) = dG (vn−p+2 ) = · · · = dG (vn ) = 1.
For i = 1, 2, . . . , n − p, setting ai = dG (vi ) and
pi =
n − dG (vi ) − p − 1
(n − p)(n − p − 1) − 2m + p
(13)
On the first Zagreb index and multiplicative Zagreb coindices of graphs
164
in (11), we get
n−p
X
i=1
−
n−p
Y
dG (vi )(n − dG (vi ) − p − 1)
−
(n − p)(n − p − 1) − 2m + p
!
dG (vi )n−dG (vi )−p−1
1
(n − p)(n − p − 1) − 2m + p
≥
i=1
(n − p)(n − ∆ − p − 1)
≥
(n − p)(n − p − 1) − 2m + p
n−p
1/(n−p)
2m − p Y
−
dG (vi )
n−p
i=1
!
(14)
that is,
(n − p − 1)(2m − p) + p − M1 (G)
−
(n − p)(n − p − 1) − 2m + p
1


n−p
Q
(n
−
p)(n
−
p
− 1) − 2m + p
dG (vi )n−dG (vi )−1 



−  i=1 n−p

Q


p
dG (vi )
≥
i=1
(n − p)(n − ∆ − p − 1)
≥
(n − p)(n − p − 1) − 2m + p
n−p
1/(n−p)
2m − p Y
dG (vi )
−
n−p
i=1
!
.
After that, we actually have
n−p
Q
dG (vi )n−dG (vi )−1
i=1
n−p
Q
"
≤
dG (vi )p
(n − p − 1)(2m − p) + p − M1 (G)
(n − p)(n − p − 1) − 2m + p
i=1
−
×
(n − p)(n − ∆ − p − 1)
(n − p)(n − p − 1) − 2m + p
n−p
1/(n−p)
2m − p Y
dG (vi )
−
n−p
i=1
!#(n−p)(n−p−1)−2m+p
.
Using the above result with (13) and the definition of Narumi-Katayama
index, we get
Y
2
(G)
=
Y
vi vj ∈E(G)
/
dG (vi ) dG (vj ) =
n
Y
dG (vi )n−dG (vi )−1
i=1
(15)
165
On the first Zagreb index and multiplicative Zagreb coindices of graphs
"
≤
p
(N K(G))
×
(n − p − 1)(2m − p) + p − M1 (G)
(n − p)(n − ∆ − p − 1)
−
(n − p)(n − p − 1) − 2m + p
(n − p)(n − p − 1) − 2m + p
2m − p
− (N K(G))1/(n−p)
n−p
#(n−p)(n−p−1)−2m+p
.
Case (ii) : The situation p = 0:
Similarly as in Case (i), for i = 1, 2, . . . , n, setting ai = dG (vi ) and pi =
n − dG (vi ) − 1
in (11), we get
n(n − 1) − 2m
"
Y
2m(n − 1) − M1 (G)
(G) ≤
2
n(n − 1) − 2m
#n(n−1)−2m
n(n − ∆ − 1)
2m
1/(n)
−
− (N K(G))
.
n(n − 1) − 2m
n
Therefore the first part of the proof is completed.
Now let us suppose that the equality holds in (12). Then all the inequalities
in the above must be equalities. For Case (i), we have p > 0 and dG (v1 ) =
dG (v2 ) = · · · = dG (vn−p ) , by Lemma 8. Hence G ∈ Γ4 . For Case (ii), we
have p = 0 and dG (v1 ) = dG (v2 ) = · · · = dG (vn ) , by Lemma 8. Hence G is
isomorphic to a regular graph.
Conversely, let G be isomorphic to a r-regular graph. Then p = 0, 2m = nr,
M1 (G) = nr2 and N K(G) = rn . We have
"
Y
2
(G) = r
n(n−r−1)
=
(n − 1)2m − nr2
n(n − ∆ − 1)
−
n(n − 1) − 2m
n(n − 1) − 2m
2m
−r
n
#n(n−1)−2m
.
Let G ∈ Γ4 . Then 2m = (n − p)∆ + p, M1 (G) = (n − p)∆2 + p and
N K(G) = ∆n−p . Now we have
"
(n − p)(n − ∆ − p − 1)
p (n − p − 1)(2m − p) + p − M1 (G)
(N K(G))
−
(n − p)(n − p − 1) − 2m + p
(n − p)(n − p − 1) − 2m + p
×
2m − p
− (N K(G))1/(n−p)
n−p
#(n−p)(n−p−1)−2m+p
= ∆p(n−p) ∆(n−p)(n−p−∆−1) = ∆(n−p)(n−∆−1) =
Y
2
(G) .
166
On the first Zagreb index and multiplicative Zagreb coindices of graphs
This completes the proof.
4
Some relations between first Zagreb index and NarumiKatayama index, and multiplicative Zagreb index and
coindices of graphs
One of the present authors compared between various topological indices in
his previous research works [8, 10, 16]. In this section we obtain some relations
among topological indices of graphs. The first example comes to our mind is
for star K1, n−1 (n ≥ 3),
Y
Y
(K1, n−1 ) ,
(K1, n−1 ) = (n − 1)n−1 > 1 =
2
2
and the second example is for a cycle Cn (n ≥ 6),
Y
Y
(Cn ) = 22n < 2n(n−3) =
(Cn ) .
2
2
Now we can compare between the second multiplicative Zagreb index and
the second multiplicative Zagreb coindex of graphs G .
Theorem 10. Let G be a graph without isolated vertices of order n with
maximum degree ∆ and minimum degree δ.! If δ ≥ n−1
( or ∆ ≤ n−1
2
2 ), then
Y
Y
Y
Y
(G) ≥
(G)
or
(G) .
(G) ≤
2
2
2
2
Proof. We clearly have
Y
2
Y
(G) =
2
(G) =
Y
dG (vi )dG (vi )
i=1
vi vj ∈E(G)
and
Y
n
Y
dG (vi ) dG (vj ) =
dG (vi ) dG (vj ) =
n
Y
dG (vi )n−dG (vi )−1 .
i=1
vi vj ∈E(G)
/
From the above, we get
Y
n
(G) Y
n−1
2
=
dG (vi )2dG (vi )−(n−1) ≥ 1 as dG (vi ) ≥ δ ≥
.
Y
2
(G) i=1
2
Similarly, one can easily prove that
Y
2
(G) ≤
Y
2
(G) if ∆ ≤
n−1
2
.
On the first Zagreb index and multiplicative Zagreb coindices of graphs
167
For the isomorphism G ∼
= K1, n−1 (n ≥ 8) , we have
Y
2
(G) +
Y
2
(G)
=
=
(n − 1)n−1 + 1 < (n − 1)2 + 2(n−1)(n−2)/2
Y
Y
(G) +
(G).
1
1
However we can present the following result:
Theorem 11. Let G be a graph of order n . If dG (vi ) ≥ 2 for all vi ∈ V (G),
then
Y
Y
Y
Y
(G) ≥
(G) +
(G).
(G) +
2
2
1
1
Proof. For each vi vj ∈
/ E(G), without loss of generality, we can assume that
dG (vi ) ≥ dG (vj ). Thus we have
dG (vi )(dG (vj ) − 1) ≥ dG (vj ) as dG (vk ) ≥ 2 for all vk ∈ V (G),
that is,
dG (vi ) dG (vj ) ≥ dG (vi ) + dG (vj ) for vi vj ∈
/ E(G).
Using the above result, we get
Y
Y
Y
(G) −
(G) =
dG (vi ) dG (vj ) −
2
1
vi vj ∈E(G)
/
≥
Y
(dG (vi ) + dG (vi ))
vi vj ∈E(G)
/
0
and
Y
2
(G) −
Y
1
(G)
Y
=
dG (vi ) dG (vj ) −
n
Y
i=1
dG (vi )dG (vi ) −
dG (vi )2
i=1
vi vj ∈E(G)
=
n
Y
n
Y
dG (vi )2 ≥ 0 as dG (vi ) ≥ 2.
i=1
From the last two results, we get the required result.
Now we obtain the relation between the first multiplicative Zagreb coindex
and the second multiplicative Zagreb coindex. Before, for simplicity, let us use
the notation A for
p(p − 1)
n(n − 1)
−m−
− p(n − p − 1)
2
2
as in Theorem 7.
168
On the first Zagreb index and multiplicative Zagreb coindices of graphs
Theorem 12. Let G be a connected graph of order n with m edges, number
of pendent vertices p, maximum degree ∆ and non-pendent minimum degree
δ. Then
p(n−p−1)
Y2
p(p−1) Y
1
4A+ 2
(i)
(G) ≥ δ + + 2
(G)
(16)
1
2
δ
with equality if and only if G is isomorphic to a regular graph or G ∈ Γ4 , and
(ii)
p(p − 1) A
∆
δ
2
+
+2
(G) ≤ 4
1
δ
∆
p(n−p−1) Y
1
× ∆+
+2
(G)
2
∆
Y2
(17)
with equality if and only if G is isomorphic to a regular graph or G ∈ Γ4 .
Proof. For dG (vi ), dG (vj ) > 1, since
s
4≤
dG (vi )
−
dG (vj )
s
dG (vj )
dG (vi )
dG (vi )
δ
∆
≥
≥
, by [4], we have
δ
dG (vj )
∆
!2
s
+4
=
≤
s
!2
dG (vi )
dG (vj )
+
≤
dG (vj )
dG (vi )
r !2
r
∆
δ
+
(18)
δ
∆
with equality holding if and only if dG (vi ) = ∆, dG (vj ) = δ for any vi vj ∈
/ E(G)
and dG (vi ) ≥ dG (vj ).
p(p − 1)
is the number
In the proof of Theorem 7, we mentioned that
2
of vertex pairs (vi , vj ) such that vi vj ∈
/ E(G) with dG (vi ) = dG (vj ) = 1 ,
p(n − p − 1) is the number of vertex pairs (vi , vj ) such that vi vj ∈
/ E(G) with
n(n − 1)
p(p − 1)
dG (vi ) = 1 or dG (vj ) = 1, and
−m−
− p(n − p − 1) is the
2
2
number of vertex pairs (vi , vj ) such that vi vj ∈
/ E(G) with dG (vi ) > 1 and
dG (vj ) > 1.
Using the above result with (18), we get
Y
Y ∆
dG (vi )
dG (vj )
δ
+
+2
≤
+
+2
dG (vj )
dG (vi )
δ
∆
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
vi vj ∈E(G)
/
=
δ
∆
+
+2
δ
∆
A
On the first Zagreb index and multiplicative Zagreb coindices of graphs
169
and
Y
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
dG (vi )
dG (vj )
+
+2
dG (vj )
dG (vi )
4A .
≥
Moreover, the above two equalities hold if and only if dG (vi ) = dG (vj ) for all
vi and vj such that dG (vi ) > 1, dG (vj ) > 1.
Now we have
Y
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
dG (vj )
dG (vi )
+
+2
dG (vj )
dG (vi )
=4
p(p − 1)
2
and
Y
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
dG (vi )
dG (vj )
+
+2
dG (vj )
dG (vi )
Y
=
vi vj ∈E(G)
/
≥
δ+
1
dG (vi ) +
+2
dG (vi )
1
+2
δ
p(n−p−1)
with equality if and only if dG (vi ) = δ for all vi such that dG (vi ) > 1.
Also we have
Y
Y dG (vi )
dG (vj )
1
+
+2
=
dG (vi ) +
+2
dG (vj )
dG (vi )
dG (vi )
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
vi vj ∈E(G)
/
≤
1
∆+
+2
∆
p(n−p−1)
with equality if and only if dG (vi ) = ∆ for all vi .
By the definition, since we have
Y
1
(G) =
Y
(dG (vi )+dG (vj )) and
vi vj ∈E(G)
/
Y
2
(G) =
Y
vi vj ∈E(G)
/
dG (vi ) dG (vj ) ,
On the first Zagreb index and multiplicative Zagreb coindices of graphs
we get
Q2
(G)
Q1
2 (G)
Y
=
vi vj ∈E(G)
/
=
170
(dG (vi ) + dG (vj ))2
dG (vi ) dG (vj )
Y Y dG (vi )
dG (vj )
dG (vi )
dG (vj )
+
+2
+
+2
dG (vj )
dG (vi )
dG (vj )
dG (vi )
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
Y dG (vi )
dG (vj )
+
+2
×
dG (vj )
dG (vi )
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
≥
δ+
1
+2
δ
p(n−p−1)
n(n − 1)
− m − p(n − p − 1)
2
4
.
Moreover, the above equality holds if and only if G is isomorphic to regular
graph (p = 0) or G ∈ Γ4 (p > 0).
Similarly,
Q2
Q1
(G)
2 (G)
=
Y Y dG (vi )
dG (vj )
dG (vi )
dG (vj )
+
+2
+
+2
dG (vj )
dG (vi )
dG (vj )
dG (vi )
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
×
Y dG (vi )
dG (vj )
+
+2
dG (vj )
dG (vi )
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
p(p − 1) p(n−p−1) A
1
∆
δ
2
≤ 4
∆+
+2
+
.
∆
δ
∆
Moreover, the above equality holds if and only if G is isomorphic to regular
graph (p = 0) or G ∈ Γ4 (p > 0). This completes the proof.
Similarly, we obtain another relation between the first multiplicative Zagreb coindex and second multiplicative Zagreb coindex. To do that, again let
us use the notation A for the algebraic statement n(n−1)
− m − p(p−1)
− p(n −
2
2
p − 1).
171
On the first Zagreb index and multiplicative Zagreb coindices of graphs
Theorem 13. Let G be a connected graph of order n with m edges, number
of pendent vertices p, maximum degree ∆ and non-pendent minimum degree
δ. Then
p(p − 1) A
2
2
(G) ≥ 2
1
∆
Y
(i)
×
1
+1
∆
p(n−p−1) Y
2
(G)
(19)
with equality if and only if G is isomorphic to a regular graph or G ∈ Γ4 , and
p(p − 1) A
2
2
(G) ≤ 2
1
δ
Y
(ii)
×
1
+1
δ
p(n−p−1) Y
2
(G)
(20)
with equality if and only if G is isomorphic to a regular graph or G ∈ Γ4 .
Proof. We have
Q
Y 1
1
1 (G)
+
=
Q
dG (vi ) dG (vj )
2 (G)
v v ∈E(G)
/
i j
=
Y
1
1
+
dG (vi ) dG (vj )
vi vj ∈E(G)
/
dG (vi )=1, dG (vj )=1
Y
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )=1
Y ×
1
1
+
dG (vi ) dG (vj )
1
1
+
dG (vi ) dG (vj )
.
vi vj ∈E(G)
/
dG (vi )>1, dG (vj )>1
We have δ ≤ dG (vi ) ≤ ∆ for non-pendent vertices. For vi vj ∈
/ E(G) with
dG (vi ) > 1, dG (vj ) > 1 ,
1
1
2
2
≤
+
≤
∆
dG (vi ) dG (vj )
δ
and for vi vj ∈
/ E(G) with dG (vi ) > 1, dG (vj ) = 1 ,
1+
1
1
1
1
≤
+
≤1+ .
∆
dG (vi ) dG (vj )
δ
On the first Zagreb index and multiplicative Zagreb coindices of graphs
172
Using the same technique as in Theorem 12, we get the required result in
(19) and (20). Moreover, the equality holds in (19) and (20) if and only if G
is isomorphic to regular graph (p = 0) or G ∈ Γ4 (p > 0). This completes the
proof.
Lemma 14. [17] Let x1 , x2 , . . . , xN be non negative numbers, and let
N
1 X
α=
xi
N i=1
and
γ=
N
Y
!1/N
xi
i=1
be their arithmetic and geometric means, respectively. Then
X √
1 X √
1
√ 2
√ 2
xi − xj ≤ α − γ ≤
xi − xj .
N (N − 1) i<j
N i<j
Moreover, equality holds if and only if x1 = x2 = · · · = xn .
Let Γ5 be the class of graphs H5 = (V, E) such that H5 is a graph of order
n with maximum degree ∆, minimum degree δ and
∆ 6= δ , dH5 (v2 ) = dH5 (v3 ) = · · · = dH5 (vn−1 ) .
u
r
tr
tr
rt
u
u
\
r
r \\ u
u
t
u
r
G5
s
sr
u
s
s
su
G6
Figure 4:
Example 15. Let G5 and G6 be the graphs depicted in Figure 4. For G5 ,
n = 9, ∆ = 7, δ = 1 and t = 3. Moreover, the degree sequence of graph G5
is (7, 2, 2, 2, 2, 2, 2, 2, 1). Hence G5 ∈ Γ5 . For G6 , n = 6, ∆ = 3, δ = 1
and t = 2. Moreover, the degree sequence of G4 is (3, 3, 3, 3, 3, 1). Hence
G6 ∈ Γ5 .
Now we obtain a relation between the first Zagreb index and NarumiKatayama index of graph G.
On the first Zagreb index and multiplicative Zagreb coindices of graphs
173
Theorem 16. Let G be a graph of order n with m edges, maximum degree ∆
and minimum degree δ. Then
(n − 2)(N K(G))2/(n−2) ≥ (∆δ)2/(n−2) (n − 3)(∆2 + δ 2 )
+ (2m − ∆ − δ)2 − (n − 3)M1 (G) (21)
and
(∆δ)2/(n−2) (2m − ∆ − δ)2 + (∆2 + δ 2 ) − M1 (G) ≥
(n − 2)(n − 3)(N K(G))2/(n−2) .
(22)
Moreover, both the above equalities hold if and only if G is isomorphic to a
regular graph or G ∈ Γ5 .
Proof. Setting in Lemma 14, N = n − 2 and xi = dG (vi )2 , i = 2, 3, . . . , n − 1,
we get
α=
γ=
N
n−1
1 X
1
1 X
xi =
dG (vi )2 =
M1 (G) − ∆2 − δ 2
N i=1
n − 2 i=2
n−2
N
Y
!1/N
xi
=
i=1
n−1
Y
!1/(n−2)
2
dG (vi )
Qn
=
i=2
dG (vi )
∆δ
i=1
2/(n−2)
2/(n−2)
=
(N K(G))
(∆δ)2/(n−2)
and
X
√
xi −
√ 2
xj
2≤i<j≤n−1
=
X
(dG (vi ) − dG (vj ))2
2≤i<j≤n−1
= (n − 3)
n−1
X
i=2
dG (vi )2 − 2
X
dG (vi ) dG (vj )
2≤i<j≤n−1
= (n − 3)(M1 (G) − ∆2 − δ 2 ) −
n−1
X
dG (vi )(2m − dG (vi ) − ∆ − δ)
i=2
= (n − 2)M1 (G) − (n − 2)(∆2 + δ 2 ) − (2m − ∆ − δ)2 .
On the first Zagreb index and multiplicative Zagreb coindices of graphs
174
Using the above results in Lemma 14, we get
(n−2)M1 (G)−(n−2)(∆2 +δ 2 )−(2m−∆−δ)2
(n−2)(n−3)
≤
≤
M1 (G)−∆2 −δ 2
n−2
−
(N K(G))2/(n−2)
(∆δ)2/(n−2)
(n−2)M1 (G)−(n−2)(∆2 +δ 2 )−(2m−∆−δ)2
&, ,
n−2
(23)
that is,
2/(n−2)
≥ (∆δ)
(n − 2)(N K(G))2/(n−2)
(n − 3)(∆2 + δ 2 ) + (2m − ∆ − δ)2 − (n − 3)M1 (G)
and
(∆δ)2/(n−2) (2m − ∆ − δ)2 + (∆2 + δ 2 ) − M1 (G)
≥ (n − 2)(n − 3)(N K(G))2/(n−2) .
By Lemma 14, one can see easily that both the equality holds in (23) if and
only if dG (v2 ) = dG (v3 ) = · · · = dG (vn−1 ) . First part of the proof is done.
First suppose that the equality holds in (21). Then dG (v2 ) = dG (v3 ) =
· · · = dG (vn−1 ) . If ∆ = δ, then G is isomorphic to a regular graph. Otherwise,
∆ 6= δ and hence G ∈ Γ5 . Conversely, one can see easily that the equality
holds in (21) for regular graph.
Let G ∈ Γ5 . Then we have ∆ 6= δ and dG (v2 ) = dG (v3 ) = · · · =
dG (vn−1 ) = d, (say). Now,
(n − 2)(N K(G))2/(n−2) = (n − 2) d2 (∆ δ)2/(n−2) ,
(n − 2)(n − 3)(N K(G))2/(n−2) = (n − 3)(n − 2) d2 (∆ δ)2/(n−2) ,
(∆ δ)2/(n−2) (2m − ∆ − δ)2 + (∆2 + δ 2 ) − M1 (G) = (n − 3)(n − 2) d2 (∆ δ)2/(n−2) ,
and (∆ δ)2/(n−2) (n − 3)(∆2 + δ 2 ) + (2m − ∆ − δ)2 − (n − 3)M1 (G)
= (∆ δ)2/(n−2) (n − 2)2 d2 − (n − 3)(n − 2) d2 = (n − 2) d2 (∆ δ)2/(n−2) .
Hence the equality holds in (21).
Similarly, we can show that the equality holds in (22) if and only if G is
isomorphic to a regular graph or G ∈ Γ5 .
Acknowledgement. The first author is supported by the National Research
Foundation funded by the Korean government with the grant no.
On the first Zagreb index and multiplicative Zagreb coindices of graphs
175
2013R1A1A2009341. The other authors are partially supported by Research
Project Offices of N. Erbakan, Uludag and Selcuk Universities. This paper
has been prepared during the Kinkar Ch. Das’s visit in Turkey that was partially funded by TUBITAK 2221-Programme. The fifth author is supported
by Uludag University Research Fund, Project Nos: F-2015/23 and F-2015/17.
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Kinkar Ch. DAS,
Department of Mathematics,
Sungkyunkwan University,
Suwon 440-746, Republic of Korea.
Email: kinkardas2003@googlemail.com
Nihat AKGUNES,
Department of Mathematics-Computer Sciences,
Necmettin Erbakan University, Faculty of Science,
Meram Yeniyol, 42100, Konya, Turkey.
Email: nakgunes@konya.edu.tr
Muge TOGAN, Aysun YURTTAS, Ismail Naci CANGUL
Department of Mathematics,
Uludag University, Faculty of Science and Art,
Gorukle Campus, 16059, Bursa, Turkey.
Emails: mugecapkin@yahoo.com.tr, aysunyurttas@gmail.com, cangul@uludag.edu.tr
Ahmet Sinan CEVIK,
Department of Mathematics,
Selcuk University, Faculty of Science,
Campus, 42075, Konya, Turkey.
Email: sinan.cevik@selcuk.edu.tr