CONSTRUCTION OF GOLDEN GRAPH-IIshodhganga.inflibnet.ac.in/bitstream/10603/31844/11... · GRAPH-II 5.1 INTRODUCTION: In this chapter we have constructed some more golden graphs. First

CHAPTER 5

CONSTRUCTION OF GOLDEN

GRAPH-II

5.1 INTRODUCTION:

In this chapter we have constructed some more golden graphs. First we have

proved logically that for which n (i.e. number of vertices) tree nU (single

headed snake) and Prism nI are golden graphs. Next which Mobious ladder

are golden graphs and 15 KC k as golden graphs. And also for which value of

kji ,, the tree ],,[ kjiT is golden graph .Similarly, for which vales of niii ,......,, 21

the tree ],.....,[ 21 niiiT is golden graphs. We have proved logically that the tree

nA (double headed snake) is not golden graph. We have proved the

graph 21 GG , where 1G is regular graph and 2G is prism as golden graphs and

also 4PKn as golden graphs. In the end We have constructed golden graphs

using the prism, Mobious ladder, trees ( ],,[ kjiT , ],.....,[ 21 niiiT ), 15 KC k , 4PG

and 5CG as we done in the chapter 4.

5.2 EXISTING RESULTS:

Definition 5.1[19]: The Mobious ladder nMG is the graph with n2 vertices

n2,......2,1 in which following pairs of vertices are adjacent:

12,.......2,1),1,( niii

)2,1( n

.,.......2,1),,( ninii

Theorem 5.2[19]: The spectrum of nMG Mobious ladder is

njjn

j

i 2,.....,1,)1(cos2

Theorem 5.3[19]: The characteristic polynomial of the complete product of

regular graphs 1G and 2G is given by the relation:

2121

21

1121 )).((

)).((

),().,(),( nnrr

rr

GGGG

Corollary 5.4[19]: Let uv be the edge whose end points are the vertices u and

w .For the graph G , let )(uv denote the sets of all circuits Z containing u or

uv , respectively. Then

vu vZ

xZGxvuGxuGxxG' )(

, )),((2),(),(.),(

)(

)),((2),(),(.),(vZ

xZGxvuGxuvGxxG

Where )(ZG is the graph obtained from G by removing the vertices

belonging to Z (in the first sum of the first formula the summation goes over all

vertices ,u adjacent to v ).

Theorem 5.5[19]:Let nr ,........,, 21 be the spectrum of the graph G , r

being the index of G . G is regular if and only if

rn

n

i

i 1

21 ………….(1)

If (1) holds, then G is regular of degree r

Theorem 5.6[19]: The number of components of a regular graph G is equal to

the multiplicity of its index.

5.3 MAIN THEOREMS

Definition 5.7: Let nUG be the tree (Single headed snake) 2n vertices as

show in figure 5.1.

FIGURE 5.1

Theorem 5.8: The graph nUG with 2n vertices is not golden graph.

Proof: Let nZG be graph with 2n vertices .

The spectra of G is

22

12cos.2

n

k, nk ,,,,,2,1,0 and zero.

We claim that G does not have golden ratio 2

51 as an eigen value.

Suppose 2

51

22

12cos.2

n

k

4

51

22

12cos

n

k, We know that

4

5136cos

l

n

k2

22

12 , Il

36222

12

l

n

k

5

12

22

12

l

n

k

)1()1(102510k nnl

0 l , because l being the number of full rotation and

representing only half of the rotation must be zero, the above equation becomes

1 2 3 n

22510k n

2

310

kn

Which a contradiction is as n is an integer

Hence claimed.

Example 5.1:

FIGURE 5.2

The spectra of the graph [figure 5.2] is -1.1442, -1.4142, 0, 1.4142, 1.4142

Definition 5.9: Let nVG be the tree (Double headed snake) with

4n vertices as show in figure 5.3.

FIGURE 5.3

Theorem 5.10: Let nVG be graph with 4n vertices is a golden graph,if

15 kn .

Proof: Let nAG be graph with 4n vertices.

The spectra of G is the union of the spectra of 4C and the path nP .[19,page 77]

The characteristic polynomial of G is given by

),(),(),( 4 xPxCxG n

We know that nP is golden graph if and only if 15 kn .

Therefore G has spectra of 4C and the path nP with 15 kn .

Hence G is golden graph , if 15 kn .

Example 5.2:

1 2 3 n

FIGURE 5.4

The spectra of the graph [figure 5.4] is -2, -1.6180, -0.6180, 0, 0, 0.6180,

1.6180, 2.

Theorem 5.11: Let nIG be a prism with vertices 3,2 kkn and is a

golden graph, if 10mod0n .

Proof: Let nIG be a prism with vertices 3,2 kkn .

The spectrum of G is

k

i2cos21 , ki ,,,,,,2,1

Since 2

2n

kkn .

We claim that in 10

Suppose 2

514cos21

n

i

2

154cos2

n

i

4

154cos

n

i

We know that4

1572cos

.

7224

ln

i

5

22

4 l

n

i

)210(20i ln

0 l , because l being the number of full rotation and representing

only half of the rotation must be zero, the above equation becomes

220i n

10mod0 n

Hence claimed

Example 5.3:

FIGURE 5.5

A spectrum of this prism [figure 5.5] is 3, 1, 1.6180, 1.6180, -0.3819, -0.3819, -

0.6180, -0.6180, -2.6180, -2.6180.

Definition 5.12: Let G be graph obtained by taking k copies of a prism nI

with vertices 1,10 kkn and attaching each copy of prism nI of a vertex to an

isolated vertex u as shown below in figure 5.6.

FIGURE 5.6

Theorem 5.13: Let G be graph as shown in figure 5.6, then G is golden

graph.

Proof: Characteristic polynomial of G is given by

u

w

k copies

11

2 ),().,(.),(.

),(),(.),(

k

nn

k

Vw

xIxIkxIx

xwuGxuGxxG

Where 1

nI is the graph obtained from nI by deleting vertex w adjacent to u

In the above expression first term is divisible by 12 xx (by theorem 5.11)

and so also second term have Characteristic polynomial of golden prism

which is divisible by 12 xx .

Therefore ),( xG is divisible by 12 xx

Hence G is golden graph .

Example 5.4:

FIGURE 5.7

The spectra of the graph [figure 5.7] is -2.7945, -2.6180, -2.6180, -1.1918, -

0.6180, -0.6180, -0.6180, -0.4947, -0.3820, -0.3820, -0.3820, 0.3655, 1,

1.1357, 1.6180, 1.6180, 1.6180, 1.8972, 3, 3.0827 .

Theorem 5.14: Let 1G and 2G be regular graphs order 1n and 2n respectively.

2G be golden prism , then 21 GG is golden graph.

Proof: Let 1G be any graph with order 1n , regularity 1r and 2G be golden

prism with order kn 101 and regularity 32 r .

The characteristic polynomial of 21 GG is given by relation

211

1

2121 )3).((

)3).((

),().,(),( nnxrx

xrx

xGxGxGG

211

1

11

121)3).((

)3).((

.().........).(3()....().........).((2

1 nnxrxxrx

xxxxxrx nn

21111

12 )3).((.()..........()....().........( 21nnxrxxxxx nn

In the above equation , the second term is polynomial of golden prism ,so

),( 21 xGG is divisible by )1( 2 xx .

Hence 21 GG is golden graph.

Example.5.5:

FIGURE 5.8

The spectra of the graph [figure 5.8] is 9.5887, -4.5887, -1.6180, -2.0636, -

1.7785, -1.7785, -1.6180, -0.2429, -0.2429, 1.2195, 1.2195, 1, 0.6180, 0.6180,

0.6675 .

Definition 5.15 : Let G be graph obtained by taking 4P and attaching to

vertices of degree 2 to every vertex of nK as shown below in figure []

FIGURE 5.9: 4PKn

Theorem 5.16: Let G be graph as in figure 5.9, then G is golden graph.

Proof: Let G be graph with n vertices and m size

FIGURE.5.10

1uG G

1u

2u

3u

4nu

1v

2v

3v

4v

.

.

.

2u

3u

4nu

2v

3v

4v

.

.

.

1v

2v

3v

4v

.

.

.

1u

2u

3u

4nu

FIGURE.5.11

The following are the observations,

There is one cycle iC of length 3 containing 1u , viz: 1321 uvvu and

There are 5n cycles iC of length 4 containing 1u , viz, 1321: uvuvuC ii , where

4,.......,3,2 ni

31 :C)( nKGV & 4i :C)( nKGV .

Thus by recurrence relation we have,

w

n

i

i xCGxwvGxvGxxG4

1

),(2),(),(.),(

),(5),(2),(2),(.),( 434,11 xKnxKxKxxuGxxG nnn

4325

1 .52))4((.2),(. nnn xnxnxxxxuGx4324

1 )5(2.2))4((2),(. nnn xnxnxxxuGx

4342

1 )5(2.2).4(22),(. nnnn xnxxnxxuGx

432

1 )5(2)4(222),(. nnn xnnxxxuGx

432

1 222),(. nnn xxxxuGx

432

1 222),(. nnn xxxxuGx

21 vuG

2u

3u

4nu

3v

4v

.

.

.

12),(. 24

1 xxxxuGx n

By inductive hypothesis ),( 1 xuG is divisible by 12 xx .

Therefore ),( xG is divisible by 12 xx .

Hence G is golden graph.

Example.5.6:

FIGURE.5.12

The spectra of the graph [figure 5.12] is -2.1926, -1.6180, 0, 0, 0, 0.6180,

3.1926

Definition 5.17: Let G be a tree ),,( kjiT , where integers are &, kji shown in

the figure 5.13.

FIGURE 5.13

Theorem 5.18: The graph ),,( kjiTG is golden graph , if

15&15,15 sksjsi .

Proof: The Characteristic polynomial of G is given by relation

),().,().,(),().,().,(

),().,().,(),().,().,(.

),(),().,(

),(),().,(),(),().,(),().,().,(.

),(),(.),(

251515152515

151525151515

1

11

xPxPxPxPxPxP

xPxPxPxPxPxPx

xPxPxP

xPxPxPxPxPxPxPxPxPx

xwuGxuGxxG

ssssss

ssssss

kji

kjikjikji

uw

. . . .

.

.

. .

. .

i

j

k

u

w

In this equation first term is divisible by 12 xx as 15 sP is golden graph iff

15 sn and also second term, third term & also fourth term.

Therefore ),( xG is divisible by 12 xx only if 15&15,15 sksjsi

Hence claimed.

Example 5.7:

FIGURE 5.14

The spectra of the graph [figure 5.14] is -1.9021, -1.1756, 1.9021, 0, 1.1756, -

1.6180, -1.6180, -0.6180, -0.6180, 1.6180, 1.6180, 0.6180, 0.6180

Definition 5.19:Let ),,( kjiT be the graph with vertices 1

,......,, 21 nuuu and G

be the graph obtained by taking k copies of ),,( kjiT and a vertex 1u of

each copy of ),,( kjiT attached to an isolated vertex u as shown in the

figure 5.15.

.

FIGURE 5.15

Theorem 5.20:Let G be graph as shown in figure 5.15 , then G golden

graph.

Proof: Characteristic polynomial of G given by

uu

xuuGxuGxxG1

),(),(.),( 1

1)),,,(().,().,().,()),,,((.

k

kji

kxkjiTxPxPxPkxkjiTx

By Theorem 5.18 , first term is a polynomial of golden graph, since ),,( kjiT

where 15&15,15 sksjsi is divisible by )1( 2 xx and also second

term.

Therefore ),( xG is divisible by )1( 2 xx

k copies

u i i

i

j

i j k j

j

j

k

k

k

1u

1u

1u1u


Example.5.8:

FIGURE 5.16

The spectra of the graph [figure 5.16] is -2.4142, -2.1010, -1.6180, -1.6180, -1.6180, -

1.6180, -1.4142, -1.2593, -0.6180, -0.6180, -0.6180, -0.6180, -0.6180, 0, 0.4142,

0.6180, 0.6180, 0.6180, 0.6180, 1.2593, 1.4142, 1.6180,

1.6180, 1.6180, 1.6180, 2.1010, 2.4142.

Definition 5.21 : Let H be any graph with vertices 1

,.....,, 21 nuuu . Let G be the

graph obtained by taking k copies of H and one ),,( kjiT and attaching the

vertex 1u of each copy of H and the vertex of tree T to an isolated vertex u

as shown in the figure 5.17.

FIGURE.5.17

Theorem 5.22: Let G be graph as shown in figure 5.17, then G golden

graph.


k copies

u H H

H

i j k

1u

1u1u

1

),(),(.),( 1

uu

xuuGxuGxxG

),(.),().],,,[(

),().,().,().,(),().],,,[(.

1

1

1

xHxHxkjiTk

xHxPxPxPxHxkjiTx

k

k

kji

k

By the Theorem 5.18 , first term is a polynomial of golden graph, since

),,( kjiT where 15&15,15 sksjsi is divisible by )1( 2 xx and also

third term. Second term polynomial of ),( xPi (0f path) which is also divisible

by )1( 2 xx .

Therefore ),( xG is divisible by )1( 2 xx .


Example.5.9:

FIGURE 5.18

The spectra of the graph [figure 5.18] is -2.3055, -1.6180, -1.6180, -1.5379,

-1.1972, -1, -0.6180, -0.6180, -0.1657, 0.4346, 0.6180, 0.6180, 1.3408,

1.6180, 1.6180, 2.0480, 2.3828

Theorem 5.23: Let nMG be Mobious ladder graph, G is golden if

4 for 4

5ofmultilpleisk

kn .

Proof: Let G be graph of order n and size m .

Then the spectrum of G is njn

j j

j 2,.....2,1,)1(cos2

Suppose 2

51)1(cos2

j

n

j

when 1,- 2

51

2cos2 evenisk

n

j

2

51

2cos2

n

j

4

51

2cos

n

j

We know that 4

5172cos

7222

ln

j

5

210

2

l

n

j

nll )420(5

0 l , because l being the number of full rotation and representing

only half of the rotation must be zero, the above equation becomes

nl 45

5

4

kn

Hence claimed

Example 5.10:

5

22

2 l

n

j

FIGURE.5.19

The spectra of the graph [figure 5.19] is -3, -1.6180, -1.6180, -0.6180,

0.6180, 0.6180, 1.6180, 1.6180, 3.

Definition 5.24: Let G be the graph obtained by attaching k copies of

nMG Mobious ladder where 4 for 4

5ofmultilpleisk

kn to an isolated

vertex u as shown in figure 5.20.

FIGURE.5.20

Theorem 5.25: Let G be graph as shown in the figure 5.20, then G golden

graph.


Vw

xwuGxuGxxG ),(),(.),(

111 )),(.()),((),(. kn

k

n

k

n xMxMkxMx

Where nM 1 is the graph obtained from nM by deleting the vertex w adjacent

to u .

5k/4vertices

5k/4vertices

5k/4vertices

5k/4vertices

u

w

k copies

By Theorem 5.23, first term is a polynomial of golden graph, since nMG is

divisible by )1( 2 xx and also second term.

Therefore, ),( xG is divisible by )1( 2 xx .

Hence G is a golden graph.

Example 5.11:

FIGURE.5.21

The spectra of the graph [figure 5.21] is -3.0370, -3, 3.0370, 3, -1.7522,

1.7522, -0.8404, 0.8404, 0, 1.6180, 1.6180, 1.6180, -1.6180, -1.6180, -

1.6180, -0.6180, -0.6180, -0.6180, 0.6180, 0.6180, 0.6180

Theorem 5.26: The graph 15 KCG k (Wheel Graph) is golden graph.

Proof: Let G be a graph of order n and size m , let kCG 51 and 12 KG of

order 21 & nn , regularity 1,2 21 rr respectively.

FIGURE.5.22

Therefore 21 GGG

Characteristic polynomial G is given by the relation

212

2

21 )2).(()2).((

),().,(),(

21nnxrx

xrx

xGxGxP GG

212

2

11

122)2).((

)2).((

.().........).(()....().........).(2(2

1 nnxrxxrx

xxrxxxx nn

21

2)2.(

)2.(

()....().........).(2(1 nnxx

xx

xxxx n

. . .

. .

5k vertices

212 )2.()....().........(1

nnxxxx n

In the above equation, the first term is polynomial of golden Cycle of order k5

,so ),(21

xP GG is divisible by )1( 2 xx .

Hence 21 GG is golden graph.

Example.5.12:

FIGURE 5.23

The spectra of the graph [figure 5.23] is -2.3166 , -2, -1.6180, -1.6180, -

0.6180, -0.6180, 0.6180, 0.6180, 1.6180, 1.6180, 4.3166

Definition 5.27: Let G be the graph obtained by attaching k copies of

12 KC k (Wheel Graph) to an isolated vertex u as shown in figure 5.24.

FIGURE 5.24

Theroem 5.28: Let G be graph as shown in the figure 5.24 , then G golden

graph.


Vw


),(.)),((),(. 5

1

1515 xCxKCkxKCx k

k

k

k

k

By theorem 5.26 , first term is a polynomial of golden graph, since

),( 15 xKC k is divisible by )1( 2 xx and also second term.

.

.

. .

.

.

5k . .

. .

5k

5k

5k

. .

. .

.

.

. k copies

u w

Therefore G is a golden graph.

Example 5.13:

FIGURE 5.25

The spectra of the graph [figure 5.25] is 4.3371, 4.3166, -2.5971, -2.3166, -2, -

1.9342, 1.4487, -1.4230, 0.1610, 0.5297, -0.5222, -1.6180, 1.6180, 1.6180,

1.6180, 0.6180, 0.6180, -1.6180, -1.6180, -0.6180, -0.6180, -0.6180, 0.6180

Definition 5.29:Let H be any graph with vertices 1

,.......,, 21 nuuu and let G

be the graph obtained by taking k copies of graph H and one copy of

12 KC k and attaching the vertex 1u of each copy of H and the vertex of

degree k5 of 12 KC k to an isolated vertex u as shown in the figure 5.26.

FIGURE 5.26

Theorem 5.30:Let G be graph as shown in figure 5.26 , then G golden

graph.


Vw

xuuGxuGxxG ),(),(.),( 1

),(.),().,().1(

),(.),(),(.),(.

15

1

1

515

xKCxHxHk

xCxHxHxKCx

k

k

k

kk

k

.

.

. .

H

H H

5k vertices

k copies

u

1u

1u1u

Where graph 1H is the graph obtained from H by deleting the vertex 1u

adjacent to u

By Theorem 5.26, first term is a characteristic polynomial of golden graph

12 KC k , since ),( 15 xKC k is divisible by )1( 2 xx and also third term.

Second term contains polynomial of ),( 5 xC k is divisible by )1( 2 xx as

kC5 is golden graph.

Therefore, ),( xG is divisible by )1( 2 xx


Example 5.14:

FIGURE 5.27

The spectra of the graph [figure 5.27] is 1, 4.3166, -2.3166, -2, 1, 1, 1.6180,

1.6180, -1.6180, -1.6180, -0.6180, -0.6180, 0.6180, 0.6180, -1.

Definition 5.31LetG betree ),........,,( 21 niiiT , integers are ,........,, 21 niii shown in the

figure 5.38.

FIGURE 5.28

Theorem 5.32:Let G be a tree ),........,,( 21 niiiT , integers are ,........,, 21 niii ,

G [as in figure 5.28]is golden graph if 1-5s,........,15,15 21 nisisi


Vw


),(...).........,( -......

..............),(...).........,(),(...).........,(.

1

1

1

11

xPxP

xPxPxPxPx

n

nn

ii

iiii

By the Theorem , first term is a polynomial of golden graph, since

),........,,( 21 niiiT where 1-5s,........,15,15 21 nisisi is divisible by

)1( 2 xx , second term and also thn term.

1i

2i

. .

. . .

.

.

. . ni



Example 5.15:

FIGURE 5.29

The spectra of figure[] is -2.3028, -1.6180, -1.6180, -1.6180, -1.3028, -

0.6180, -0.6180, -0.6180, 0 , 0.6180, 0.6180, 0.6180, 1.3028, 1.6180,

1.6180, 1.6180, 2.3028 .

Definition 5.33:Let G be the graph obtained by attaching k copies

of ),........,,( 21 niiiT to an isolated vertex u as shown in figure 5.30..

FIGURE 5.30

Theorem 5.34:Let G be graph as shown in the figure 5.30 , then G

golden graph.


Vw


)),,........,,(().,(...).........,()),,........,,((. 1

21121 1

k

nii

k

n xiiiTxPxPkxiiiTxn

1i 2i ni

. . . . .

w

.

. . . .

.

.

.

. . .

.

.

.

.

u

1i

2i

ni

1i2i

2i

ni

ni

1i

k

copies

By the Theorem 5.32, first term is a polynomial of golden graph, since


)1( 2 xx and also second term.



Example 5.16:

FIGURE 5.31

The spectra of the graph [figure 5.31] is 0, -1.7321 , -1.0000, 0.1226, 0.1226,

1.7549 , 1.7321, -1.6180, -0.6180, 1.6180, 0.6180 , -1.7321 , 1.0000, -

1.0000, 1.7321 ,-0.0000, 0.0000, 1.0000, 1.0000, 1.0000. 1.0000, 0.0000, -

0.0000, -1.5321, -0.3473, 1.8794, -1.7321, -1.7321, -1.0000, -1.0000, -0.0000,

1.7321, 1.7321, 1.0000, 1.0000, 1.0000, 1.0000, 0.0000, 0.0000, 1.0000,

1.0000, 1.0000, 1.0000 .

Definition 5.35: Let H be any graph with vertices 1

,.....,, 21 nuuu and let G be

the graph obtained by taking k copies graph H and one tree

),........,,( 21 niiiT and attaching the vertex 1u of each copy of H and one vertex

of ),........,,( 21 niiiT to an isolated vertex u as shown in figure 5.32

FIGURE 5.32

Theorem 5.36: Let G be graph as shown in the figure 5.32, then G golden

graph.


Vw


1i 2i

. . . . . .

ni

1u

1u

H

H

H u

k copies

1u

ni

1

11

1

121

,().,(.),(...).........,(k

,(.),(...).........,(,(.)),,........,,((.

1

1

k

ii

k

ii

k

n

xHxHxPxP

xHxPxPxHxiiiTx

n

n

Where 1H is the graph obtained from H by deleting the vertex 1u adjacent

to u .

By the Theorem , first term is a polynomial of golden graph, since


)1( 2 xx . Second term and third term consists of paths which are golden

graphs .



Example 5.17:

FIGURE 5.33

The spectra of the graph[figure 5.33] is -2.4995, -1.6180, -1.6180, -1.6180,

-1.5314, -1.2391, -1, -0.6180, -0.6180, -0.6180, -0.1399, 0.4166, 0.6180,

0.6180, 0.6180, 1.3593, 1.6180, 1.6180, 1.6180, 2.1007, 2.5334,

Definition 5.37: Let 1G be any graph and 41 PG be the graph with vertices

1,.....,, 21 nuuu . Let G be the graph obtained by taking k copies of 41 PG and

attaching the vertex 1u of each copy of 41 PG to an isolated vertex u as

shown in the figure

FIGURE 5.34

Theorem 5.38: Let G be the graph shown in the figure 5.34, then G is

golden graph.


uu

xuuGxuGxxG1

),(),(.),( 1

1

41241 )().,(.),(. kkPGxGkxPGx …………..1

Where 2G is the graph obtained from 41 PG by deleting vertex 1u adjacent

tou .

In the equation (1), the first term the characteristic polynomial of

)( 41 PG is divisible by 12 xx because 41 PG is golden graph and also

the second term.

k copies

u

4PG

4PG 4PG

4PG

1u

1u

1u

1u

Therefore, ),( xG is divisible by 12 xx .


Example 5.18:

FIGURE 5.35

The eigenvalues of the graph [figure 5.35] are -2.1365, -5.8255, -1.6180, -

1.6180, -1.5440, -1, -1, -1, -0.4296, -0.4179, 0.6180, 0.6180, 0.7968, 5.2434,

5.3134.

Definition 5.39 : Let 1G be any graph and 51 CG be the graph with

vertices 1

,.....,, 21 nuuu . Let G be the graph obtained by taking k copies of

51 CG and attaching the vertex 1u of each copy of 51 CG to an isolated

vertex u as shown in the figure 5.36.

FIGURE 5.36

Theorem 5.40: Let G be the graph shown in the figure 5.36, then G is

golden graph.


uu

xuuGxuGxxG1

),(),(.),( 1

1u

1u

1u 1u

k copies

u

5CG

5CG

5CG

5CG

1

51251 )().,(.),(. kkCGxGkxCGx …………..(1a)

Where 2G is the graph obtained from 51 CG by deleting vertex 1u adjacent

to u .

In the equation (1a), the first term the characteristic polynomial of

)( 51 CG is divisible by 12 xx because 51 CG is golden graph and

also the second term.

Therefore, ),( xG is divisible by 12 xx .


Example 5.19:

FIGURE 5.37

The eigenvalues of the graph [figure 5.37] are -2.2184, -1.8730, -1.8033, -

1.6180, -1.6180, -1.6180, -1, -1, -1, -1, 0.2486, 0.6180, 0.6180, 0.6180, 1.3617,

5.8730, 5.9085.

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CONSTRUCTION OF GOLDEN GRAPH-IIshodhganga.inflibnet.ac.in/bitstream/10603/31844/11... · GRAPH-II 5.1 INTRODUCTION: In this chapter we have constructed some more golden graphs. First