Some results on Some results on

sum graphsum graphWei JianxinWei Jianxin

School of Mathematics and Information ofSchool of Mathematics and Information of

Lu Dong UniversityLu Dong University

Email:wjx0426@tom.comEmail:wjx0426@tom.com

1.Introduction1.Introduction The notion of sum graph was introduced The notion of sum graph was introduced by F.Harary. by F.Harary.

The sum graph The sum graph GG(+) (+) (S)(S) of a finite subset S of a finite subset S of the set Nof the set N(*)(*) of all positive integers is the of all positive integers is the graph (S,E) with uv graph (S,E) with uv ∈∈E if and only if u+vE if and only if u+v∈∈ S.S.

A graph G is said to be a sum graph if it is A graph G is said to be a sum graph if it is isomorphic to the sum graph isomorphic to the sum graph GG(+) (+) (S)(S) of soof some Sme SNN(*) (*) .In this case,we say that S gives .In this case,we say that S gives a sum labelling for G.a sum labelling for G.

1.Introduction1.Introduction The sum number (G) is the

smallest number of isolated vertices which when added to G result in a sum

graph. In 1994,F,Harary introduced the concepts of integral sum graph and integral sum number of a graph with SZ (the set of all integers) insteadof SSNN(*)(*).

1.Introduction1.Introduction

In a labelling of a sum graph G,vertics whose label corresponds to an edge of G are said to be working vertices.

Graphs that can only belabelled in such a way that allthe working vertics are also isolates are called exclusive.

1.Introduction1.Introduction

From a practical point of view,sum graph labelling can be used as a compressed representation of a graph,a data structure for representing the graph,and an alternative method for defining and storing graphs.

1.Introduction1.Introduction Now the research aims at two aspects.

One is to study the relation between the (integral)sum number and other parameters and structure of graph.

The other is at determining the sum number of some graph classes.

2.The sum number of crown 2.The sum number of crown Theorem 2.1 Let n Theorem 2.1 Let n 3,then 3,then (C(Cnn☉☉ K K11)= 1)= 1

12

3

58

13

21

34

55

2268

31

7330

°75

Sum labeling of(CSum labeling of(C77☉☉ K K1 1 )) K K11

2.The sum number of crown2.The sum number of crownTheorem 2.2 Let n Theorem 2.2 Let n 3,then 3,then (C(Cnn☉☉ K K11))(*) (*) =1=1

1

2

3

5

813

21

34

55

89

144

°233

90

124

137

142

Sum labeling of(CSum labeling of(C55☉☉ K K1 1 ))(*) (*) K K11

2.The sum number of crown2.The sum number of crownTheorem2.3 Let n Theorem2.3 Let n 3,then 3,then (S(C(S(Cnn☉☉ K K11))=1))=1

1 23

81321

3455

35

48

53

144

589

197

245°280

Sum labeling of S(CSum labeling of S(C44☉☉ K K1 1 )) K K11

3.The sum number of all 2-regular grap3.The sum number of all 2-regular graphshs

Theorem3.1 Theorem3.1 The sum number of all 2-regular grapThe sum number of all 2-regular grap

hs with the exception of hs with the exception of CC44 is 2. is 2.

1

3

6

8

4

14

9

41

23

18

50

45

68

95

163

258

321

°389°579

Sum labeling of (CSum labeling of (C44CC44CC44CC55))KK11

4.The sum number of K4.The sum number of K11••••••1r1r

Theorem 4.1 Let r Theorem 4.1 Let r 3, then 3, then (K(K11r11r)=r.)=r.

Theorem 4.2 Let k>3,r Theorem 4.2 Let k>3,r 2k-3, then 2k-3, then (K(K11••••••1r1r) =r+) =r+

( k-1),where k is the number of 1 in K( k-1),where k is the number of 1 in K1 1 •••••• 1r . 1r .1 11 21

12 22 32 42 52

23

33

43

53

63

73Sum labeling of (KSum labeling of (K11151115))7K7K11

13

5.The sum number of wind-mill5.The sum number of wind-mill

Theorem 5.1 Theorem 5.1 (D(Dnn)=2,)=2,(D(Dnn)=0,where D)=0,where Dn n is the Dutis the Dutch n-windmill.ch n-windmill.

1

10 19

11

18

12

1713

16

15

14

7

-7

-5

5 3-3

-1

1

9 -9

Sum labeling of (DSum labeling of (D55))2K2K11 and and Integral sum labeling of DIntegral sum labeling of D55

°20°29

5.The sum number of wind-mill5.The sum number of wind-mill

Theorem 5.2 FTheorem 5.2 Fmm is exclusive,where F is exclusive,where Fm m is Frenchis French

m-windmill. m-windmill.

FF44

6.Miscellaneous Results6.Miscellaneous Results

Theorem 6.1Theorem 6.1 Let S Let S11 , S , S2 2 denote the sum labelling denote the sum labelling

s of s of GG(G)K(G)K11,,HH(H)K(H)K11 respectively and suppos respectively and suppos

e maxSe maxS11=m,minS=m,minS22 =n,(m,n)=1,if minS =n,(m,n)=1,if minS11/ {n} / {n} 2n o 2n o

r 2max Sr 2max S22/ {m} / {m} m,then m,then (G (G H) H) (G)(G)+ + (H)(H) -1. -1.

6.Miscellaneous Results6.Miscellaneous Results

Theorem 6.2Theorem 6.2 Let G be a connected integral sum graph with n(3) vertices and m edges .If G has only one

saturated vertex ,then (n-1) m

Theorem 6.2Theorem 6.2 The smallest order of an integral sum graph with clique number (G)( 3) is 2(G)-3.

Theorem 6.3Theorem 6.3 The smallest order of a unit sum graph with clique number (G)( 3) is 2(G)-2.

23 31

8

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