Gen. Math. Notes, Vol. 23, No. 2, August 2014, pp.85-95
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Weighted Szeged Index of Generalized
Hierarchical Product of Graphs
S. Nagarajan1 , K. Pattabiraman2 and M. Chandrasekharan3
1
Department of Mathematics, Kongu Arts and Science College
Erode - 638 107, India
E-mail: profnagarajan@rediffmail.com
2
Department of Mathematics, Annamalai University
Annamalainagar - 608 002, India
E-mail: pramank@gmail.com
3
Department of Mathematics, Erode Arts and Science College
Erode - 638 009, India
E-mail: mc.brinda@gmail.com
(Received: 16-4-14 / Accepted: 26-5-14)
Abstract
The Szeged index of a graph G, denoted by S z(G) =
nGu (e)nGv (e). Simiuv=e∈E(G)
P larly, the Weighted Szeged index of a graph G, denoted by S zw (G) =
dG (u)+
uv=e∈E(G)
dG (v) nGu (e)nGv (e), where dG (u) is the degree of the vertex u in G. In this paper, the
exact formulae for the weighted Szeged indices of generalized hierarchical product
and Cartesian product of two graphs are obtained.
Keywords: Generalized hierarchical product, Cartesian product, Szeged index,
weighted Szeged index.
1
P
Introduction
All the graphs considered in this paper are connected and simple. A vertex x ∈
V(G) is said to be equidistant from the edge e = uv of G if dG (u, x) = dG (v, x),
where dG (u, x) denotes the distance between u and x in G. The edges e = uv
and f = xy of G are said to be equidistant edges if min {dG (u, x), dG (u, y)} =
86
S. Nagarajan et al.
min {dG (v, x), dG (v, y)}. The degree of the vertex u in G is denoted by dG (u).
For an edge uv = e ∈ E(G), the number of vertices of G whose distance to
the vertex u is smaller than the distance to the vertex v in G is denoted by nGu (e);
analogously, nGv (e) is the number of vertices of G whose distance to the vertex v in
G is smaller than the distance to the vertex u; the vertices equidistant from both the
ends of the edge e = uv are not counted.
Graph theory successfully provides the chemists with a variety of very useful
tools, namely, different topological indices. A topological index of a graph is a
parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. In theoretical chemistry, molecular structure descriptors (also
called topological indices) are used for modeling physicochemical, pharmacologic,
toxicologic, biological and other properties of chemical compounds [14]. Several
types of such indices exist, especially those based on vertex and edge distances.
One of the most intensively studied topological indices is the Wiener index.
The two topological indices, namely, the Szeged index of G, denoted by S z(G),
and, weighted Szeged index of G, denoted by S zw (G), are defined as follows:
P
S z(G) =
nGu (e)nGv (e),
e = uv ∈ E(G)
P
S zw (G) =
dG (u) + dG (v) nGu (e)nGv (e).
e = uv ∈ E(G)
A graph G with a specified vertex subset U ⊆ V(G) is denoted by G(U). Barriere
et al. [1, 2] defined a new product of graphs, namely, the generalized hierarchical
product, as follows: Let G and H be two graphs with a nonempty vertex subset
U ⊆ V(G). Then the generalized hierarchical product, denoted by G(U) u H, is the
graph with vertex set V(G) × V(H) and two vertices (g, h) and (g0 , h0 ) are adjacent
if and only if g = g0 ∈ U and hh0 ∈ E(H) or, gg0 ∈ E(G) and h = h0 (see Fig.1).
The Cartesian product, G H, of graphs G and H has the vertex set V(G H) =
V(G)×V(H) and (u, x)(v, y) is an edge of G H if u = v and xy ∈ E(H) or, uv ∈ E(G)
and x = y, see Fig.2.
Fig.1. P5 (U) u P4 , where U = {2, 3, 4}
To each vertex u ∈ V(G), there is an isomorphic copy of H in G H and to each
vertex v ∈ V(H), there is an isomorphic copy of G in G H. But in the generalized
Weighted Szeged Index of Generalized...
87
hierarchical product, to each vertex u ∈ U, there is an isomorphic copy of H and to
each vertex v ∈ V(G), there is an isomorphic copy of G. In particular, if U = V(G),
then G H = G(U) u H.
Fig.2. P5 P4
Let G and H be simple connected graphs with vertex sets V(G) = {u1 , u2 , . . . , un }
and V(H) = {v1 , v2 , . . . , vm } , respectively, and let U be a nonempty subset of G.
Then V(G(U) u H) = V(G) × V(H); for our convenience, we write V(G(U) u H) =
Sn
Sm
i=1 Xi , wherenXio= {ui } × V(H); we may also write V(G(U) u H) =
j=1 Y j , where
Y j = V(G) × v j . We denote the vertices of Xi by {(ui , v j ) |1 ≤ j ≤ m} and the
vertices of Y j by {(ui , v j )|1 ≤ i ≤ n} and we call Xi , 1 ≤ i ≤ n, the i-th layer of
G(U) u H and Y j , 1 ≤ j ≤ m, the j-th column of G(U) u H.
Weighted Szeged index of graph G has been introduced by Illić and Milosavljević [11]. They are given the upper and lower bounds for weighted vertex PI index
of graph. Also the exact formula for weighted vertex PI index of Cartesian product
of graphs is obtained in [11]. The Szeged index studied by Gutman [5], Gutman
and Dobrynin [6] and Khadikar et. al. [9] is closely related to the Wiener index of
a graph. Basic properties of Szeged index and its analogy to the Wiener index are
discussed by Klavžar et. al.[8] . It is proved that for a tree T the Wiener index of
T is equal to its Szeged index. Ashrafi et. al. [10] have explained the differences
between Szeged and Wiener indices of graphs. The mathematical properties and
chemical applications of Szeged index are well studied by Dobrynin et. al. [3],
Gutman et. al. [4] and Randic et. al. [13]. Recently Pisanski and Randic [12]
studied the measuring network bipartivity using Szeged index. In this paper, the
exact formulae for the weighted Szeged indices of generalized hierarchical product
and Cartesian product of two graphs are obtained.
2
Weighted Szeged Index of G(U) u H
Let G = (V, E) be a graph and U ⊆ V. In G(U), an u - v path through U is an uv path in G containing some vertex w ∈ U(vertex w could be the vertex u or v).
Let dG(U) (u, v) denote the length of a shortest u-v path through U in G. Note that,
if one of the vertices u and v belongs to U, then dG(U) (u, v) = dG (u, v). A vertex
x ∈ V(G(U)) is said to be equidistant from e = uv ∈ E(G(U)) through U in G(U),
88
S. Nagarajan et al.
if dG(U) (u, x) = dG(U) (v, x). For an edge e in G(U), let NG(U) (e) denote the number
of equidistant vertices of e through U in G(U). Then S z(G(U)) and S zw (G(U)) are
defined as follows:
S z(G(U)) =
X
nG(U)
(e) + nG(U)
(e)
u
v
e = uv ∈ E(G)
S zw (G(U)) =
X
dG(U) (u) + dG(U) (v) nG(U)
(e) + nG(U)
(e) .
u
v
e = uv ∈ E(G)
For an edge e = uv ∈ E(G), let TG (e, u) be the set of vertices closer to u than v and
TG (e, v) be the set of vertices closer to v than u. That is,
TG (e, u) = {x ∈ V(G)|dG (u, x) < dG (v, x)}
TG (e, v) = {x ∈ V(G)|dG (u, x) > dG (v, x)}.
Similarly, for an edge e = uv ∈ E(G(U)),
TG(U) (e, u) = {x ∈ V(G(U))|dG(U) (u, x) < dG(U) (v, x)}
TG(U) (e, v) = {x ∈ V(G(U))|dG(U) (u, x) > dG(U) (v, x)}.
The proof of the following lemma is left to the reader as it follows easily from
the structure of G(U) u H. The lemma is used in the proof of the main theorem of
this section.
Lemma 2.1. Let G and H be graphs with U ⊆ V(G). Then
(i) |V(G(U) u H)| = |V(G)| |V(H)| , |E(G(U) u H)| = |E(G)| |V(H)| + |E(H)| |U| .
(ii) The degree of the vertex (g, h) ∈ V(G(U) u H) is dG(U) (g) + φU (g)dH (h), where
φU denote the characteristic 
function on the set U which is 1 on U and 0 outside U.


dG(U) (g, g0 ) + dH (h, h0 ), i f h , h0 ,
(iii) dG(U) u H ((g, h)(g0 , h0 )) = 

dG (g, g0 ), i f h = h0 .
Next we compute the weighted Szeged index of the generalized hierarchical
product of two connected graphs G and H.
Theorem 2.2. Let G and H be two connected graphs with n, m vertices and p, q
edges and let U be a nonempty
subset of V(G). Then S zw (G(U) u H) =
P
2
2
2n S z(H)
dG(U) (ur ) +n |U| S zw (H)+mS zw (G)+m(m−1)2 S zw (G(U))+m(m−
ur ∈ U
P P G(U)
1)
dG(U) (ui )+dG(U) (uk ) nGui (e)nG(U)
(e)nGuk (e) +2q
φU (ui )+
uk (e)+nui
ui uk ∈ E(G)
ui uk ∈ E(G)
P
G(U)
G
φU (uk ) nGui (e)nGuk (e)+2q(m−1)
φU (ui )+φU (uk ) nGui (e)nG(U)
(e)+n
(e)n
(e)
uk
ui
uk
ui uk ∈ E(G)
G(U)
P
+ 2q(m − 1)2
φU (ui ) + φU (uk ) nui (e)nG(U)
uk (e).
ui uk ∈ E(G)
Weighted Szeged Index of Generalized...
89
Proof. Let V(G) = {u1 , u2 , . . . , un }, V(H) = {v1 , v2 , . . . , vm } and let U be a
nonempty subset of V(G). For our convenience, we partition the edge set of G(U) u
H into two sets, E1 = {(ur , vi )(ur , vk ) | ur ∈ U, vi vk ∈ E(H)} and E2 = {(ur , vi )(u s , vi ) |
ur u s ∈ E(G), vi ∈ V(H)}, that is, E1 = ∪ui ∈U E(hXi i) and E2 = ∪mj=1 E(hY j i).
Let e = vi vk ∈ E(H) and let v j ∈ T H (e; vi ). Then, for any ur ∈ U and e0 ∈
E1 ⊂ E(G(U) u H), the distance of (ur , vi ) to each vertex of Y j , is less than its
distance to the vertex (ur , vk ) in G(U) u H. It can be observed that if some vertex
v s < T H (e, vi ), then all the vertices of the column Y s are not in TG(U)uH (e0 ; (ur , vi )) in
G(U) u H. Also if vr is equidistant to e in H, then every vertex of Yr is equidistant
to e0 . Consequently, for the edge e0 ∈ E1 ( of G(U) u H) corresponding to e ( in H),
0
H
nG(U)uH
(ur ,vi ) (e ) = n nvi (e)
(1)
0
H
nG(U)uH
(ur ,vk ) (e ) = n nvk (e).
(2)
and similarly,
Hence for E1 defined as above,
X
0 G(U)uH 0
dG(U)uH ((ur , vi )) + dG(U)uH ((ur , vk )) nG(U)uH
(ur ,vi ) (e )n(ur ,vk ) (e )
(ur ,vi )(ur ,vk ) = e0 ∈ E1
X
=
dG(U) (ur ) + dH (vi ) + dG(U) (ur ) + dH (vk ) n2 nvHi (e) nvHk (e) ,
(ur ,vi )(ur ,vk ) = e0 ∈ E1
by (1) and (2), where e = vi vk ∈ E(H),
X
X
= n2
2dG(U) (ur ) nvHi (e)nvHk (e)
ur ∈ U vi vk = e ∈ E(H)
+n2
X
X
dH (vi ) + dH (vk ) nvHi (e)nvHk (e) , since |E1 | = |U| |E(H)| ,
ur ∈ U vi vk = e ∈ E(H)
X
= 2n S z(H)
dG(U) (ur ) + n2 |U| S zw (H).
2
(3)
ur ∈ U
Let e = ui uk ∈ E(G(U)). Then, for any v` ∈ V(H) e0 = (ui , v` )(uk , v` ) ∈ E2 ⊂
E(G(U)u H). If u j ∈ TG (e, ui ) then (u j , v` ) ∈ TG(U)uH (e0 , (ui , v` )). Hence {(u j , v` )|u j ∈
TG (e, ui )} ⊆ TG(U)uH (e0 , (ui , vr )). If ` , s then since dG(U)uH ((ui , v` ), (u j , v s )) <
dG(U)uH ((uk , v` ), (u j , v s )) if and only if dG(U) ((ui , v j ) + dH (v` , v s ) < dG(U) ((uk , v j ) +
dH (v` , v s ) if and only if dG(U) ((ui , v j ) < dG(U) ((uk , v j ).
Therefore {(u j , v s )|u j ∈ TG(U) (e, ui )} ⊆ TG(U)uH (e0 , (ui , v` )).
Hence,
TG(U)uH (e0 , (ui , v` )) = {(u j , v` )|u j ∈ TG (e, ui )} + {(u j , v s )|u j ∈ TG(U) (e, ui )} .
Consequently,
0
G
G(U)
nG(U)uH
(e)
(ui ,v` ) (e ) = nui (e) + (m − 1)nui
(4)
90
S. Nagarajan et al.
and similarly,
0
G
G(U)
nG(U)uH
(uk ,v` ) (e ) = nuk (e) + (m − 1)nuk (e).
(5)
Hence for E2 defined as above
X
(ui ,v` )(uk ,v` ) = e0 ∈ E2
X
=
X
0 G(U)uH 0
dG(U)uH ((ui , v` )) + dG(U)uH ((uk , v` )) nG(U)uH
(e
)
n
(e
)
(ui ,v` )
(uk ,v` )
G(U)
dG(U) (ui ) + dG(U) (uk ) nG
(e)
ui (e) + (m − 1)nui
v` ∈V(H) ui uk ∈ E(G)
nG
uk (e)
+ (m −
X
X
+
1)nG(U)
uk (e)
!
G(U)
dH (v` ) φU (ui ) + φU (uk ) nG
(e)
ui (e) + (m − 1)nui
v` ∈V(H) ui uk ∈ E(G)
nG
uk (e)
+ (m −
1)nG(U)
uk (e)
!
,
by (4) and (5), where e = ui uk ∈ E(G(U))
= S 1 + S 2 , where S 1 and S 2 are the sums of the above terms, in order.
(6)
We shall calculate S 1 and S 2 of (6) separately.
S1 =
X
X
G
dG(U) (ui ) + dG(U) (uk ) nG
ui (e)nuk (e)
v` ∈V(H) ui uk ∈ E(G)
X
+(m − 1)
X
G(U)
G(U)
G
dG(U) (ui ) + dG(U) (uk ) nG
(e)n
(e)
+
n
(e)n
(e)
uk
ui
ui
uk
v` ∈V(H) ui uk ∈ E(G)
X
+(m − 1)
2
X
dG(U) (ui ) + dG(U) (uk ) nG(U)
(e)nG(U)
ui
uk (e)
v` ∈V(H) ui uk ∈ E(G)
= m
X
G
dG (ui ) + dG (uk ) nG
ui (e)nuk (e)
ui uk ∈ E(G)
X
+m(m − 1)
G(U)
G(U)
dG(U) (ui ) + dG(U) (uk ) nG
(e)nG
ui (e)nuk (e) + nui
uk (e)
ui uk ∈ E(G)
X
+m(m − 1)
2
dG(U) (ui ) + dG(U) (uk ) nG(U)
(e)nG(U)
ui
uk (e)
ui uk ∈ E(G)
2
= mS zw (G) + m(m
S zw (G(U))
X− 1)
G(U)
G(U)
+m(m − 1)
dG(U) (ui ) + dG(U) (uk ) nG
(e)nG
ui (e)nuk (e) + nui
uk (e) .
(7)
ui uk ∈ E(G)
S2 =
X
X
G
dH (v` ) φU (ui ) + φU (uk ) nG
ui (e)nuk (e)
v` ∈V(H) ui uk ∈ E(G)
X
+ (m − 1)
X
G(U)
G(U)
dH (v` ) φU (ui ) + φU (uk ) nG
(e)nG
ui (e)nuk (e) + nui
uk (e)
v` ∈V(H) ui uk ∈ E(G)
+ (m − 1)2
X
X
v` ∈V(H) ui uk ∈ E(G)
dH (v` ) φU (ui ) + φU (uk ) nG(U)
(e)nG(U)
ui
uk (e)
(8)
91
Weighted Szeged Index of Generalized...
Using (6) and the sums S 1 and S 2 in (7) and (8), respectively, we have,
S zw (G(U) u H)
X
= 2n2 S z(H)
dG(U) (ur ) + n2 |U| S zw (H) + mS zw (G) + m(m − 1)2 S zw (G(U))
ur ∈ U
X
+m(m − 1)
G(U)
G(U)
G
dG(U) (ui ) + dG(U) (uk ) nG
(e)
+
n
(e)n
(e)
(e)n
u
u
ui
uk
i
k
ui uk ∈ E(G)
+2q
X
G
φU (ui ) + φU (uk ) nG
ui (e)nuk (e)
ui uk ∈ E(G)
X
+2q(m − 1)
G(U)
G(U)
φU (ui ) + φU (uk ) nG
(e)nG
ui (e)nuk (e) + nui
uk (e)
ui uk ∈ E(G)
+2q(m − 1)2
X
φU (ui ) + φU (uk ) nG(U)
(e)nG(U)
ui
uk (e).
ui uk ∈ E(G)
In the above theorem, if we set U = V(G), we obtain the following corollary.
Corollary 2.3. [11] Let G and H be connected graphs. Then S zw (G H) =
|V(H)|3 S zw (G) + |V(G)|3 S zw (H) + 4 |V(H)|2 |E(H)| S z(G) + 4 |V(G)|2 |E(G)| S z(H).
Let G1 , G2 , . . . , Gn be graphs with vertex set V(Gi ) and edge set E(Gi ), 1 ≤
n
e
i ≤ n. Denote by Gi the Cartesian product of graphs G1 , G2 , . . . , Gn . Clearly,
i=1
n
n
n
n
e
Q
e
Q
V( Gi ) = |V(Gi )| . By induction on n, one can see that E( Gi ) = |V(Gi )|
i=1
i=1
i=1
i=1
n
3
n
n
n
e
P
P
Q
|E(Gi )|
V(G ) .
. In [8], S, Klavžar et al. have proved S z( G ) = S z(G )
i=1
i
|V(Gi )|
i=1
i
i=1
j
j=1, j,i
Next we compute a similar result for the weighted Szeged index.
n
e
Theorem 2.4. Let G1 , G2 , . . . , Gn be connected graphs. Then S zw ( Gi ) =
i=1
3
2 Q
n
n
n
n
P
Q
P
V(G j ) + 4
S zw (Gi )
S z(Gi ) V(G j ) E(G j )
|V(Gk )|3 .
i=1
j=1, j,i
i, j=1 i, j
k=1,i,k, j
Proof. The case n = 2 was proven in Theorem 2.2. We continue our argument by
92
S. Nagarajan et al.
mathematical induction. Suppose that the results is valid for some n graphs.
S zw
n+1
m
n
m
G i = S zw
Gi Gn+1
i=1
i=1
3
n
n
m
m
Gi
Gi S zw (Gn+1 ) + |V(Gn+1 )|3 S zw
= V
i=1
i=1
2 n
n
n
m
m m 2
Gi
Gi S z(Gn+1 ) + |V(Gn+1 )| |E(Gn+1 )| S z
Gi E
+4 V
i=1
i=1
i=1
= S zw (Gn+1 )
n
Y
|V(Gi )| + |V(Gn+1 )|
3
3
n
X
n
Y
3
V(G j )
S zw (Gi )
i=1
j=1, j,i
i=1
n
X
+4
2 S z(Gi ) V(G j ) E(G j )
i, j=1 i, j
+4 S z(Gn+1 )
n
X
|V(Gi )|2 |E(Gi )|
i=1
n
X
=
S zw (Gi )
i=1
+
2 S z(Gi ) V(G j ) E(G j )
i≤ j≤n
=
n
X
i=1
V(G )3 +
j
j=1, j,i
n
X
2 3
V(G j ) + 4
S z(Gi ) V(G j ) E(G j )
j=1, j,i
X
k=1,i,k, j
n
Y
n
Y
3 V(G j )
S z(Gi )
i=1
n+1
Y
|V(Gk )|3
j=1, j,i
|V(Gn+1 )| |E(Gn+1 )|
n+1
X
n
Y
i, j=1, i, j
n+1
Y
n+1
Y
|V(Gk )|3
k=1, i,k, j
|V(Gk )|3
k=1, i,k, j
n
n
Y
X
3
2 V(G j ) + 4
S zw (Gi )
S z(Gi ) V(G j ) E(G j )
j=1, j,i
i, j=1 i, j
n
Y
|V(Gk )|3 .
k=1,i,k, j
The proof of the following corollary directly follows from Theorem 2.4.
n
e
e
Corollary 2.5. Let G be a connected graph. Then S zw ( Gn ) = S zw ( G) =
i=1
3n−4 n |V(G)|
|V(G)| S zw (G) + 4(n − 1) |E(G)| S z(G) .
Example 2.6. SupposeeQn denotes the hypercube of dimension n. Then by Theorem 2.4, S zw (Qn ) = S zw ( K2n ) = n2 2(3n−2) .
Let us consider the graph G whose vertices are the N−tuples b1 b2 . . . bN with
bi ∈ {0, 1, . . . , ni − 1}, ni ≥ 2, and two vertices be adjacent if the corresponding
tuples differ in precisely one place; such a graph is called a Hamming graph. It is
well-known fact that a graph G is a Hamming graph if and only if it can be written
93
Weighted Szeged Index of Generalized...
in the form G =
N
e
Kni and so the Hamming graph is usually denoted by Hn1 n2 ...nN . In
i=1
the following lemma, the weighted Szeged index of a Hamming graph is computed.
It is easy to see that S z(Kn ) = n(n−1)
and S zw (Kn ) = n(n − 1)2 . The proof of the
2
following lemma follows from Theorem 2.4.
Lemma 2.7. Let G be a Hamming graph with!above parameter. Then
2
N N
N
P
P
(ni −1)(n j −1) Q 3
1 − n1i +
ni .
S zw (Hn1 n2 ...nN ) =
n2
i=1
i, j=1, i, j
i
i=1
Let Cn and Pn denote the cycle and path on n vertices, respectively. It is known
2
3
that S z(Cn ) = n4 when n is even, and n(n−1)
otherwise and S z(Pn ) = ( n+1
3 ); see [7].
4
3
It can be easily verified that S zw (Cn ) = n when n is even, and n(n − 1)2 otherwise
2
and S zw (Pn ) = 2(n−1)(n3 +n−3) .
Using Theorems 2.2, 2.4 and S zw (Pn ),S zw (Cn ),S z(Pn ) and S z(Cn ), we obtain
the exact weighted Szeged indices of the following graphs.
Example 2.8. The graphs Ln = Pn K2 , R = Pn Cm , S = Cm Cn and T = Pm Pn
are known as ladder, C4 nanotubes, C4 nanotorus and grid, respectively. The exact
weigted Szeged indices of these graphs are given below.
1. S zw (Ln ) = 14n3 − 4n2 − 24n + 16.
3


 m3 (10n3 − 3n2 − 10n + 6) if m is even
2. S zw (R) =

 2m3 (n3 + n2 − 5n + 3) + m(m − 1)2 n2 (2n − 1) if m is odd.
 3


2n3 m(m − 1)2 + 2m3 n(n − 1)2 if m is odd n is odd






2n3 m(2m2 − 2m + 1) if m is odd n is even
3. S zw (S )= 


2m3 n(2n2 − 2n + 1) if m is even n is odd





4m3 n3 if m is even n is even.
2
2m2 (n−1)
(2mn2 +2mn−3m−n2 −n)+ 2n (m−1)
(2nm2 +2mn−3n−m2 −m).
3
3
 k

2Q 3


k
n
if each ni is even


 i=1 i
5. S zw (Cn1 Cn2 . . . Cnk ) = 
k
k

Q
P
2



1 − n1i
if each ni is odd.
k n3i
i=1
i=1



e k
k2 n3k if each ni is even
each ni = n, then S zw ( Cn ) = 

k2 (n − 1)2 n3k−2 if each ni is odd.
4. S zw (T ) =
If
Example 2.9. Let G = Cn1 Cn2 . . . Cnk and H = Cm1 Cm2 . . . Cmr , where
ni , 1 ≤ i ≤ k are even and m j , 1 ≤ j ≤ r are odd. Using Theorem 2.2 and the above
example we obtain the weighted Szeged index of the graph GH.
Q
Q
k
r
P
S zw (GH) =
n3i
m3j k2 + kr + (k + r) r(1 − m1i )2 .
i=1
j=1
i=1
e k
e r
If each ni = n ≥ 3 is even
and m j = m ≥ 3 is odd,
then G = Cn and H = Cm
and S zw (GH) = n3k m3r k2 + kr + (k + r)r(1 − m1 )2 .
94
S. Nagarajan et al.
Example 2.10. Using Theorem 2.4, we obtain the exact weighted Szeged index
of the grid graph Pn1 Pn2 . . . Pnk .
Q
P
k
k (n −1)(n2 +n −3)
k
P
i
i
i
S zw (Pn1 Pn2 . . . Pnk ) = 32
n3i
+
(1 − n1i )(1 + n1i )(1 −
3
ni
i=1
i=1
i, j=1, i, j
1
).
nj
e
3(k−1)
If each ni = n, then S zw ( Pkn ) = 2k(n−1)n
(n2 + n − 3) + (k − 1)(n + 1) .
3
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