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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
Hence G is golden graph.
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.
Proof: Characteristic polynomial of G given by
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 .
Hence G is golden graph.
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.
Proof: Characteristic polynomial of G given by
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.
Proof: Characteristic polynomial of G given by
Vw
xwuGxuGxxG ),(),(.),(
),(.)),((),(. 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.
Proof: Characteristic polynomial of G given by
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
Hence G is a golden graph.
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
Proof: Characteristic polynomial of G given by
Vw
xwuGxuGxxG ),(),(.),(
),(...).........,( -......
..............),(...).........,(),(...).........,(.
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
Therefore ),( xG is divisible by )1( 2 xx
Hence G is a golden graph.
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.
Proof: Characteristic polynomial of G given by
Vw
xwuGxuGxxG ),(),(.),(
)),,........,,(().,(...).........,()),,........,,((. 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
),........,,( 21 niiiT where 1-5s,........,15,15 21 nisisi 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 5.16:
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.
Proof: Characteristic polynomial of G given by
Vw
xwuGxuGxxG ),(),(.),(
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
),........,,( 21 niiiT where 1-5s,........,15,15 21 nisisi is divisible by
)1( 2 xx . Second term and third term consists of paths which are golden
graphs .
Therefore ),( xG is divisible by )1( 2 xx
Hence G is a golden graph.
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.
Proof: Characteristic polynomial of G given by
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 .
Hence G is golden graph.
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.
Proof: Characteristic polynomial of G given by
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 .
Hence G is golden graph.
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.