Filomat 26:3 (2012), 467–472
DOI 10.2298/FIL1203467Y
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
The Szeged, vertex PI, first and second Zagreb indices of corona
product of graphs
Zahra Yarahmadi
a
, Ali Reza Ashrafi
b
a
Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad, University, Khorramabad, I.R. Iran
b
Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan, I.R. Iran
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 193955746, Tehran, I.R. Iran
Abstract. The corona product GoH of two graphs G and H is defined as the graph obtained by taking one
copy of G and |V(G)| copies of H and joining the i-th vertex of G to every vertex in the i−th copy of H. In
this paper, the Szeged, vertex PI and the first and second Zagreb indices of corona product of graphs are
computed.
1. Introduction
Let G be a connected graph with vertex and edge sets V(G) and E(G), respectively. The distance between
the vertices u and v of G is denoted by d
G
(u, v) and it is defined as the number of edges in a shortest path
connecting the vertices u and v. A topological index is a numerical quantity related to a graph which is
invariant under graph automorphisms. One of the most famous topological indices is the Wiener index
introduced by Harold Wiener [25] as an aid to determining the boiling point of paraffin. Since then, the
index has been shown to correlate with a host of other properties of molecules (viewed as graphs). For
more information about the Wiener index in chemistry and mathematics see [4 − 6, 8 − 11]. The Wiener
index of G is the sum of distances between all unordered pairs of vertices of G, W(G) =
{u,v}⊆V(G)
d
G
(u, v).
The Szeged index Sz(G) is another topological index was introduced by Ivan Gutman [9]. It is defined as
Sz(G) =
e=uv∈E(G)
n
u
(e|G)n
v
(e|G), where n
u
(e|G) is the number of vertices of G lying closer to u than v and
n
u
(e|G) is defined analogously, see [1, 2, 18, 20] for mathematical properties and chemical meaning of this
topological index. It is a well-known fact that for an acyclic graph T, Sz(T) = W(T). The vertex PI index is a
recently introduced topological index defined as, PI
v
(G) =
e=uv∈E(G)
[n
u
(e|G) + n
v
(e|G)], [1, 17]. Notice that
for computing Szeged and vertex PI indices, vertices equidistant from u and v are not taken into account. In
general, if G is a bipartite graph then PI
v
(G) = |V(G)||E(G)|. This shows that the vertex PI index is the same
for bipartite graphs with n vertices and q edges. On the other hand, the vertex PI index of bipartite graphs
has the maximum value between graphs with exactly n vertices and q edges. Finally, the first and second
Zagreb indices are defined as M
1
(G) =
u∈V(G)
deg
2
G
u and M
2
(G) =
e=uv∈E(G)
deg
G
u deg
G
v, respectively,
where de1
G
u is the degree of vertex u in G. The interested readers for more information on Zagreb indices
can be referred to [12, 13, 16].
2010 Mathematics Subject Classification. Primary 05C76; Secondary 05C12, 05C07
Keywords. Corona product, Wiener index, Szeged index, vertex PI index, first and second Zagreb index
Received: 26 July 2010; Accepted: 31 November 2010
Communicated by Dragan Stevanovi
´
c
Email addresses: z.yarahmadi@gmail.com (Zahra Yarahmadi), ashrafi@kashanu.ac.ir (Ali Reza Ashrafi)
Z. Yarahmadi, A.R. Ashrafi / Filomat 26:3 (2012), 467–472 468
Graph operations play an important role in the study of graph decompositions into isomorphic sub-
graphs. Let G and H be two simple graphs. If |V(G)| = n and |E(G)| = q, we say that G is an (n, q)−graph.
We also say that G is of order n. The corona product GoH of two graphs G and H is an important graph
operation defined as the graph obtained by taking one copy of G and |V(G)| copies of H and joining the
i−th vertex of G to every vertex in i−th copy of H. If G is an (n, q)−graph and H is an (n, q)−graph then
|V(GoH)| = n + nn
′
and |E(GoH)| = q + nq
′
+ nn
′
. The i−th copy of H is denoted by H
i
, 1 ≤ i ≤ n as shown
in Fig. 1. It is clear from the definition that corona product of two graphs is not commutative. Obviously,
GoH is connected if and only if G is connected. Also if H contains at least one edge then GoH is not bipartite
graph.
G
H
1
H
2
H
n
.
. .
. .
.
Figure 1: The corona product of two graphs
In this paper we study some topological indices of a graph under corona product. We encourage the
reader to consult [3, 15] for our notation and [7, 14, 19 − 24] for more information on graph operations under
some topological indices.
2. Main Results
In this section some topological indices of corona product of two graphs are computed. We start by
computing the Szeged index of corona product. In what follows, the number of triangles containing an
edge e = uv is denoted by t
uv
.
Theorem 2.1. Let G be a connected graph of order n. For every (m, q)-graph H, the Szeged index of GoH is given by
Sz(GoH) = nM
2
(H) + n
e=uv∈E(H)
t
uv
(t
uv
− deg
H
u − deg
H
v) + (m + 1)
2
Sz(G) + mn(mn + n − 1) − 2nq.
Proof. By definition of Szeged index,
Sz(GoH) =
e=uv∈E(GoH)
n
u
(e|GoH)n
v
(e|GoH).
We partition the edges of GoH in to three subset E
1
, E
2
and E
3
, as follows:
E
1
= {e ∈ E(GoH) | e ∈ E(H
i
) , 1 ≤ i ≤ n},
E
2
= {e ∈ E(GoH) | e ∈ E(G)},
E
3
= {e ∈ E(GoH) | e = uv , u ∈ V(H
i
) , 1 ≤ i ≤ n , v ∈ V(G)}.
Therefore,
Sz(GoH) =
e∈E
1
n
u
(e
|
GoH )n
v
(e
|
GoH ) +
e∈E
2
n
u
(e
|
GoH )n
v
(e
|
GoH ) +
e∈E
3
n
u
(e
|
GoH )n
v
(e
|
GoH ).
Z. Yarahmadi, A.R. Ashrafi / Filomat 26:3 (2012), 467–472 469
For every e = uv ∈ E(H) if there exists w ∈ V(H) such that uw < E(H) and vw < E(H) then d
GoH
(u, w) =
d
GoH
(v, w) = 2. Also if there exists w ∈ V(H) such that uw ∈ E(H) and vw ∈ E(H) then d
GoH
(u, w) =
d
GoH
(v, w) = 1. Hence n
u
(e
|
GoH ) = deg
H
u − t
uv
and so
e∈E
1
n
u
(e|GoH)n
v
(e|GoH) = n
e=uv∈E(H)
(deg
H
u − t
uv
)(deg
H
v − t
uv
). (1)
We now assume that e = uv ∈ E
2
. Then for each vertex w closer to u than v, the vertices of the copy of H
attached to w are also closer to u than v. Since each copy of H has exactly m vertices, n
u
(e|GoH) = (m+ 1)n
u
(e|G).
Similarly, n
v
(e|GoH) = (m + 1)n
v
(e|G). Therefore,
e∈E
2
n
u
(e|GoH)n
v
(e|GoH) =
e∈E(G)
(m + 1)
2
n
u
(e|G)n
v
(e|G). (2)
Finally, we assume that e = uv ∈ E
3
, de1
H
u = k and {u
1
, u
2
, ..., u
k
} are adjacent vertices of u in H
i
. By definition
of corona product of graphs, v is adjacent to vertices u
1
, ..., u
k
. Thus for each j, 1 ≤ j ≤ k, u
j
is equidistant
from u and v. On the other hand, every vertex of GoH other than u, u
1
, .., u
k
are closer to v than u. This
implies that n
v
(e|GoH) = |V(GoH)| − (de1
H
u + 1) and n
u
(e|GoH) = 1. Therefore,
e∈E
3
n
u
(e|GoH)n
v
(e|GoH) =
e∈E
3
[|V(GoH)| − (de1
H
u + 1)]. (3)
We now apply Equations 1-3, we have:
Sz(GoH) = n
e=uv∈E(H)
(deg
H
u − t
uv
)(deg
H
v − t
uv
) +
e=uv∈E
2
(1 + m)
2
n
u
(e
|
G)n
v
(e
|
G)
+
e=uv∈E
3
[
|
V(GoH)
|
− (deg
H
u + 1)]
= nM
2
(H) + n
e=uv∈E(H)
t
uv
(t
uv
− deg
H
u − deg
H
v) + (m + 1)
2
Sz(G) + mn(mn + n − 1) − 2nq.
By above calculations, one can see that,
Sz(GoH) = nM
2
(H) + n
e=uv
e∈E(H)
t
uv
(t
uv
− deg
H
u − deg
H
v) + (m + 1)
2
Sz(G) + mn(mn + n − 1) − 2nq.
Corollary 2.2. Let G be a connected graph of order n and H be a triangle-free (m, q)-graph. Then,
Sz(GoH) = nM
2
(H) + (m + 1)
2
Sz(G) + mn(mn + n − 1) − 2nq.
Proof. Substitute t
uv
= 0, for every edge e = uv ∈ E(H), in the statement of Theorem 1.
Let P
n
, n ≥ 2, C
n
and S
n
denote the path, the cycle and the star on n vertices, respectively.
Corollary 2.3. The following equalities are hold:
a. Sz(P
n
oP
m
) =
1
6
n(n
2
− 1)(m + 1)
2
+ mn(mn + n + 1) − 6n m , 2
3
2
n(n
2
+ 4n − 3) m = 2
,
b. Sz(S
n
oP
m
) =
(n − 1)
2
(m + 1)
2
+ mn(mn + n + 1) − 6n m , 2
3(5n
2
− 7n + 3) m = 2
,
Z. Yarahmadi, A.R. Ashrafi / Filomat 26:3 (2012), 467–472 470
c. Sz(P
n
oC
m
) =
1
6
n(n
2
− 1)(m + 1)
2
+ mn(mn + n + 1) m , 3
8
3
n(n
2
− 1) + 6n(2n − 1) m = 3
,
d. Sz(C
n
oC
m
) =
1
4
n
3
(m + 1)
2
+ mn(mn + n + 1) 2|n, m , 3
1
4
n(n − 1)
2
(m + 1)
2
+ mn(mn + n + 1) 2 |n, m , 3
,
e. Sz(P
n
oS
m
) =
1
6
n(n
2
− 1)(m + 1)
2
+ n(m − 1)(m − 3) + mn(mn + n − 1).
Corollary 2.4. Let G = P
n
and H = K
c
m
be an empty graph of order m. Then GoH is a Caterpillar tree and
Sz(P
n
oH) =
1
6
n(n
2
− 1)(m + 1)
2
+ mn(mn + n − 1).
In the following theorem, we apply a similar reasoning as in the proof of Theorem 1 to calculate the vertex
PI index of corona product of graphs.
Theorem 2.5. Let G be a connected graph of order n and H be (m, q)-graph, then the vertex PI index of GoH is given
by
PI
v
(GoH) = (m + 1)PI
v
(G) + nM
1
(H) + n
2
m(m + 1) − 2n(q + 3t),
where t is the number triangles of H.
Proof. By definition
PI
v
(GoH) =
e∈E
1
[n
u
(e
|
GoH ) + n
v
(e
|
GoH )] +
e∈E
2
[n
u
(e
|
GoH ) + n
v
(e
|
GoH )] +
e∈E
3
[n
u
(e
|
GoH ) + n
v
(e
|
GoH )].
We compute each summation as follows:
PI
v
(GoH) = n
e=uv∈E(H)
[(deg
H
u − t
uv
) + (deg
H
v − t
uv
)] +
e=uv∈E(G)
[n
u
(e
|
G) + n
v
(e
|
G)](m + 1)
+
e=uv∈E
3
[
|
V(GoH)
|
− deg
H
u].
= n
e=uv∈E(H)
[(deg
H
u + deg
H
v) − 2n
e=uv∈E(H)
t
uv
+ (m + 1)PI
v
(G) + mn
|
V(GoH)
|
−
e∈E
3
deg
H
u.
By above calculations, PI
v
(GoH) = (m + 1)PI
v
(G) + nM
1
(H) + n
2
m(m + 1) − 2(nq + 3nt).
Corollary 2.6. Suppose H is triangle-free (m, q)-graph and G is a connected graph of order n. Then
PI
v
(GoH) = (m + 1)PI
v
(G) + nM
1
(H) + n
2
m(m + 1) − 2nq.
Corollary 2.7. The following equalities are hold:
a. PI
v
(P
n
oP
m
) = mn(mn + 2n + 1) + n(n − 5),
b. PI
v
(P
n
oS
m
) = n
2
(m + 1)
2
+ n(m − 2)
2
− 3n,
c. PI
v
(P
n
oC
m
) =
n
2
(m + 1)
2
+ n(m − 1) m , 3
4n(4n − 1) m = 3
,
d. PI
v
(C
n
oP
m
) =
mn(mn + 2n + 2) + n(n − 4) 2|n
mn(mn + 2n + 1) + n(n − 5) 2 |n
.
We end this section by computing the Zagreb indices of corona products.
Theorem 2.8. Let G be (n, q
′
)-graph and H be (m, q)-graph then
M
1
(GoH) = M
1
(G) + nM
1
(H) + 4(mq
′
+ nq) + mn(m + 1),
M
2
(GoH) = n[M
1
(H) + M
2
(H) + q] + (2q + m)(2q
′
+ mn) + mM
1
(G) + M
2
(G) + m
2
q
′
.
Z. Yarahmadi, A.R. Ashrafi / Filomat 26:3 (2012), 467–472 471
Proof. By definition,
M
1
(GoH) =
u∈V(GoH)
deg
2
GoH
u
= n
u∈V(H)
deg
2
GoH
u +
u∈V(G)
deg
2
GoH
u
= n
u∈V(H)
(deg
H
u + 1)
2
+
u∈V(G)
(deg
G
u + m)
2
= n
u∈V(H)
(deg
2
H
u + 2 deg
H
u + 1) +
u∈V(G)
(deg
2
G
u + 2m deg
G
u + m
2
)
= nM
1
(H) + 4nq + mn + M
1
(G) + 4mq
′
+ m
2
n
= M
1
(G) + nM
1
(H) + 4(mq
′
+ nq) + mn(m + 1).
In order to compute the second Zagreb index, suppose that V(G) = {v
1
, v
2
, ..., v
n
} and V(H) = {u
1
, u
2
, ..., u
m
}.
We partition of the set E(GoH) into three parts and evaluate the resulting sums:
M
2
(GoH) =
e=uv
deg
GoH
u deg
GoH
v = n
e=uv
e∈E(H)
deg
GoH
u deg
GoH
v
+
e=uv
u∈V(H)
v∈V(G)
deg
GoH
u deg
GoH
v +
e=uv
e∈E(G)
deg
GoH
u deg
GoH
v
= n
e=uv
e∈E(H)
(deg
H
u + 1)(deg
H
v + 1)
+
n
i=1
m
j=1
(deg
H
u
j
+ 1)(deg
G
v
i
+ m)
+
e=uv
e∈E(G)
(deg
G
u + m)(deg
G
v + m)
= n[
e=uv
e∈E(H)
deg
H
u deg
H
v+
e=uv
e∈E(H)
(deg
H
u + deg
H
v)+
e=uv
e∈E(H)
1]
+
n
i=1
(deg
G
v
i
+ m)
m
j=1
(deg
H
u
j
+ 1)
+
e=uv
e∈E(G)
deg
G
u deg
G
v + m
e=uv
e∈E(G)
(deg
G
u+ deg
G
v) +
e=uv
e∈E(G)
m
2
= n[M
1
(H) + M
2
(H) + q] + (2q + m)
n
i=1
(deg
G
v
i
+ m)
+ M
2
(G) + mM
1
(G) + m
2
q
′
.
From these equations,
M
2
(GoH) = n[M
1
(H) + M
2
(H) + q] + (2q + m)(2q
′
+ mn) + mM
1
(G) + M
2
(G) + m
2
q
′
which completes the proof.
