The Hyper
Zagreb Index of Graph Operations
G.H.SHIRDELa,,H.REZAPOURa AND A.M.SAYADIb
a
Department of Mathematic, Faculty of Basic Sciences, University of Qom, Qom, Iran
b
School of Mathematic, Statistics and Computer Science, University of Tehran, Tehran, Iran
(ReceivedJuly 1, 2013; Accepted November 11, 2013)
A
BSTRACTLet G be a simple connected graph. The first and secondZagreb indices have been introduced as M1(G) v V(G)degG(v)2 and M2(G) uv E(G)degG(u)degG(v), respectively, where degGv(degGu) is the degree of vertex v(u). In this paper, we define a new distance-based named HyperZagreb as HM(G)euvE(G)(degG(u)degG(v))2. In this paper, the HyperZagreb index of the Cartesian product, composition, join and disjunction of graphs are computed.
Keywords: HyperZagreb index, Zagreb index, graph operation.
1.
I
NTRODUCTIONIn this paper, it is assumed that G is a connected graph with vertex and edge sets V(G) and E(G), respectively. We consider only simple connected graphs, i.e. connected graphs without loops and multiple edges. For a graph G, the degree of a vertex v is the number of edges incident to v, denoted bydegGv. A topological index Top(G) of a graph G, is a number with this property that for every graph H isomorphic to G, Top(H) = Top(G). Usage of topological indices in chemistry began in 1947 when chemist Harold Wiener developed the most widely known topological descriptor, the Wiener index, and used it to determine physical properties of types of alkanes known as paraffin. In a graph theoretical language the Wiener index is equal to the sum of distances between all pairs of vertices of the respective graph. For a nice survey in this topic we encourage the reader to consult [14].
Numerous other indices have been defined. The Zagreb indices have been introduced more than thirty years ago by Gutman and Trinajestić [6]. They are defined as:
) (
2 1( ) deg( )
G V v
v G
M
) (
2( ) deg( )deg( ).
G V uv
v u G
M
We encourage the reader to consult [1, 5, 16, 17, 18] for historical background, computational techniques and mathematical properties of Zagreb indices. We define the HyperZagreb index as:
) (
2
)) ( deg ) ( (deg )
(
G E uv e
G
G u v
G
HM .
The Cartesian product GH of graphs G and H has the vertex set )
( ) ( )
(G H V G V H
V and (a,x)(b,y) is an edge of GH if ab and xyE(H), or abE(H) and x y.
The join GH of graphs G and H is a graph with vertex set V(G)V(H) and edge setE(G)E(H){uv:uV(G)and vV(H)}.
The composition G[H] or (GoH) of graphs G and H with disjoint vertex sets V(G) and V(H) and edge sets E(G) and E(H) is the graph with vertex set V(G)V(H) and
) , (u1 v1
u is adjacent to v(u2,v2) whenever u1 is adjacent to u2 or u1u2 and v1 is
adjacent tov2 .
The symmetric difference GH of two graphs G and H is the graph with vertex set V(G)V(H) and
} both not but ) H ( E 2 v 2 u or ) G ( E 1 v 1 u | ) 2 v , 1 v )( 2 u , 1 u {( ) H G (
E .
The tensor product GH of two graphs G and H is the graph with vertex set
) ( ) (G V H
V andE(GH){u1,u2)(v1,v2)|u1v1E(G)and u2v2E(H)}.
The corona product GH is defined as the graph obtained from Gand H by taking one copy of Gand V(G) copies of H and then joining by an edge each vertex of the ith copy of H is named (H,i) with the ith vertex of G.
2.
N
EWG
RAPHI
NVARIANTLet us begin with a few examples, then we will give a crucial lemma related to distance properties of some graph operations. Our main results are applications of this lemma in computing exact formulae for some graph operations under HyperZagreb index.
Example 1. In this example the HyperZagreb index of some well-known graphs are
calculated. We first consider the complete graph Kn. Then the degree of any vertex in this graph is n1, thus
) ( 3 2 ) 1 ( 2 )) ( deg ) ( (deg ) ( kn E uv e k kn u v n n
K HM
n
n .
Let Km,ndenote a complete bipartite graph. Then we have:
. ) K ( E uv e 2 ) n m ( mn 2 )) v ( k deg ) u ( K (deg ) n , m K ( HM n , m n , m n , m
For a cycle graph with nvertices, we have:
) ( 2 16 )) ( deg ) ( (deg ) ( n n n C E uv e C Cn u v n
C
HM .
For a path with nvertices, we have:
) ( 2 30 16 )) ( deg ) ( (deg ) ( n n n P E uv e p pn u v n
P
HM .
For wheel on n1 vertices, we have:
) 45 6 ( )) ( deg ) ( (deg ) ( 2 ) (
2
n n n v u W HM n n n W E uv e W W n .Finally, for a ladder graph with 2n vertices, n3, the HyperZagreb index can be computed by the following formula:
156 108 )) ( deg ) ( (deg ) ( ) ( 2 2 2 2
2
n v u L HM n n n L E uv e L L n .Lemma 1. Let Gand H be two connected graphs, then we have:
(a) V(GH) V(GH) V(G[H]) V(GH) V(G)V(H),
, ) ( ) ( ) ( ) ( )
(G H E G V H V G E H
E
, ) ( ) ( ) ( ) ( )
(G H E G E H V G V H
E
, ) ( ) ( ) ( ) ( ]) [
(G H E G V H 2 E G V G
E
, ) ( ) ( 2 ) ( ) ( ) ( ) ( )
(G H V G V H 2 E G V G 2 E G E H
E
. ) ( ) ( 4 ) ( ) ( ) ( ) ( )
(G H E G V H 2 E H V G 2 E G E H
E
(c) If (a,b) is a vertex of GHthen degGH
(a,b)
degG(a)degH(b).(d)If (a,b) is a vertex of G[H] then degG[H]
(a,b)
V(G)degG(a)degH(b). (e) If (a,b) is a vertex of GH orGH, we have:degGH
(a,b)
V(H)degG(a)V(G)degH(b)2degG(a)degH(b). degGH
(a,b)
V(H)degG(a)V(G)degH(b)degG(a)degH(b). (f) If a is a vertex of GH then, we have:
) ( )
( ) ( deg
) ( )
( ) ( deg )
( deg
H V a G V a
G V a H V a a
H G H
G
Proof: The parts (a) and (b) are consequence of definitions and some famous results of the
book of Imrich and Klavzar [7]. For the proof of (c-f) we refer to [8].
Theorem 1. Let G and H be graphs. Then we have:
)) G ( 1 M ) H ( V ) H ( 1 M ) G ( V ( 5 ) H ( HM ) G ( HM ) H G (
HM
+
2E(G) E(G)E(H) )
H ( V ) H ( E 2 ) G ( V 8
+ V(G)V(H)
(V(H) V(G))24(E(G) E(H)
.Proof: From the definition we know:
)} H ( V v ), G ( V u : uv { ) H ( E ) G ( E ) H G (
E U U
So, we have:
) H G ( E uv
2 ) v H G deg u H G (deg )
H G (
HM
=
) H ( E uv
2 ) v H G deg u H G (deg
+
E(G)
uv
2 ) v H G deg u H G (deg
+ .
)} H ( V v ), G ( v u : uv { uv
2 ) v H G deg u H G (deg
It is easy to see that:
( ) ( ) ( )
(1)) ( 4 ) ( )
deg
(deg 2 1
A uv
H G H
G u v HM H V G M H V G E H
( ) ( ) ( )
. (2) )( 4 ) ( )
deg
(deg 2 1
B uv
H G H
G u v HM H V H M G V H E G
Finally, we can write:
C uv
H V v
G V u
G G
H G H
G u v u V H v V G
) (
) (
2 2
) ( deg
) ( deg
) deg (deg
V(G)M1(H) V(H)M1(G)8E(G)E(H)
+
(V(H) V(G))24(E(G) E(H) )
H ( V ) G ( V
+4 V(G)2E(H) V(H)2E(G). (3)
Combining these three equations ((1), (2), (3)) will complete the proof.
Theorem 2. Let G and H be graphs. Then we have:
) ( ) ( 16 ) ( ) ( 8 ) ( ) ( 8 ) ( ) ( ) ( ) ( )
(G H V G HM H V H HM G EG M1 H E H M1G EG E H
HM
Proof: From the definition of the Cartesian product of graphs, we have:
} ), ( ),
( :
) , )( , {( )
(G H a x b y ab E G x yor xy E H a b
E
therefore we can write:
) H G ( E ) y , b )( x , a (
2 ) y , b ( H G deg ) x , a ( H G deg )
H G ( HM
+
) H ( E xy
) y , a )( x , a (
2 ) y , a ( H G deg ) x , a ( H G deg
=
) G ( E xy
) x , b )( x , a (
2 ) x , b ( H G deg ) x , a ( H G deg
+
V(G)
a xy E(H)
2 y H deg x H deg a G deg 2
=
V(H)
x ab E(G)
2 b G deg a G deg x H deg 2
This implies that,
. ) ( ) ( 16 ) ( ) ( 8 ) ( ) ( 8 ) ( ) ( ) ( )
(G HM H V H HM G E G M1 H E H M1 G E G E H
V
G[H]
V(H)4HM(G) V(G)HM(H) 2E(G)V(H)M1(H)HM
8V(H)E(H)
E(G)M1(H)2M1(G)V(H)
.Proof: From the definition of the compositionG[H] we have:
) G ( E ab ]) H [ G ( E ) y , b )( x , a ( 2 ) y , b ( ] H [ G deg ) x , a ( ] H [ G deg ] H [ G HM +
) ( ]) [ ( ) , )( , ( 2 ] [ ][ ( , ) deg ( , ) .
deg H E xy H G E y a x a H G H
G a x a y
So we have:
) ( ]) [ ( ) , )( , ( 2 ] [ ][ ( , ) deg ( , )
deg G E ab H G E y b x a H G H
G a x b y
) ( ( ) ( ) 2 ) ( deg ) ( deg ) ( ) ( deg ) ( deg ) ( H Vx yV H ab EG
H G
H
G a x V H b y
H V
) ( ( ) 1 2 2 )) ( deg ) ( )(deg ( ) ( 2 )) ( deg ) ( (deg ) ( ) ( ) ( H Vx y V H
H H
H
H x y V H M G x y
G E G HM H V ) 4 ( . ) H ( E 2 ) H ( V ) G ( 1 M 8 ) H ( E ) H ( V ) G ( E 8 ) H ( 1 M ) H ( V ) G ( E 2 ) G ( HM 4 ) H (
V
Moreover,
) ( ]) [ ( ) , )( , ( 2 ] [ ][ ( , ) deg ( , )
deg H E xy H G E y a x a H G H
G a x a y
) ( ( ) 2 ) ( deg ) ( deg ) ( deg ) ( 2 G Va xyEH
H H
G a x y
H V
) ( 1 2 2 ) ( ) ( deg ) ( 4 ) ( ) ( deg ) ( ) ( 4 G V a GG a HM H V H a M H
H V H E ) 5 ( . ) H ( E ) H ( V ) H ( 1 M 8 ) H ( HM ) G ( V 2 ) H ( V ) H ( E ) G ( 1 M
4
It is easy to see that the summation of (4) and (5) complete the proof.
Theorem 4. Let G and H be graphs. Then we have:
2 ( ) 1
) ( ) ( 4 ) ( ) ( 5 ) ( ) ( 5 ) ( ) ( ) ( )
(GH HM G V G HM H V H M1 G V G M1 H V H E G V H
HM
( ) ( )
( ) ( )
( ) 2 ( ) 4 ( )
. )(
8E H V G E G V G V H V H 3 V H E H
Proof: Let AE(G),BE(H,i) and Cedges between Gand(H,i) then:
Auv uv A
G G
H G H
G u v u V H v V H
( ) ( )
. (6) )( 4 )
(G V H E G M1 G
HM
( )
1 ( )
2 2
1 deg 1 deg )
deg (deg
G V
i uv EH
H H
B uv
H G H
G u v u v
V(G)
4E(H) HM(H)4M1(H)
. (7)
)} , ( ), ( {
2 2
1 deg )
( deg
deg deg
i H V v G V u
C uv
H G
c uv
H G H
G u v u V H v
V(G)V(H)
V(H)2 14E(H) 2V(H)
4V(H)2E(G)V(H)M1(G)V(G)M1(H)4E(H)
V(G) 2E(G)
. (8) Combining these three equations ((6), (7), (8)) complete the proof.3.
F
UTUREW
ORKIn addition, further research we seek to generalize the above theorems for n graph. Then you can see the similarities between generalization of the first and second Zagreb and their generalization to HayperZagreb index of graphs.
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