tree1
TRANSCRIPT
26VIT1984-2010
Creating Stars
On Combinatorial Identities of
Trees
N.Chandramowliswaran
Applied Algebra Division
School of Advanced Sciences
VIT University 26
VIT1984-2010
Creating Stars
Claude Berge Bill Tutte
A Place to Learn; A Chance To Grow
Let T be a Tree on ‘q’ edges.
Define represents the subtrees of the form
(or)
( where a,b are non negative integers , )
(α,β) (β,α)T T
a b
1u
2u
3u
ua
1v
vb
3v
2vu v
1u
2u
3u
ub
1v
va
3v
2vu v
3
26VIT1984-2010
Creating Stars
denotes the total number of subtrees of the form
(or)
(α,β) (β,α)N T T
(α,β) (β,α)T T
uv E(T)
d(u) 1 d(v) 1 d(u) 1 d(v) 1
a b b a
(α,β) (β,α)N T T
1u
2u
3u
ua
1v
vb
3v
2vu v
1u
2u
3u
ub
1v
va
3v
2vu v
uv E(T)
d(u) 1 d(v) 1
a a
(α,α)N TIf a b
4
26VIT1984-2010
Creating Stars
For each path length 2,
Define represents the subtrees of the form
(where a,g,b are non negative integers, )
(or)
denotes the total number of subtrees of the form
uv E(T), uw,wv E(T), d(u,v) 2
g g (α, ,β) (β, ,α)T T
1x
2x
3x
xa
zg
1y
yb
3y
2y
1z2z 3z
u vw
a b
, (α, ,β) (β, α)N T Tg g
, , (α, β) (β, α)T Tg g
1x
2x
3x
xb
zg
1y
ya
3y
2y
1z2z 3z
u vw
5
26VIT1984-2010
Creating Stars
If a b
The degree of vertex ‘u’ in the Subtree , = 1a
The degree of vertex ‘v’ in the Subtree , = 1b
The degree of vertex ‘w’ in the Subtree = 2g
Pi = Path length of i
g(α, ,β)T
(α,β)T
g(α, ,β)T
(α,β)T
g(α, ,β)T
u vw
d(u) 1 d(w) 2 d(v) 1
α γ α
(α,γ,α)
uv E(T)d(u,v)=2
N T =
d(u) 1 d(w) 2 d(v) 1
α γ βN
d(u) 1 d(w) 2 d(v) 1+
β γ α
(α,γ,β) (β,γ,α)
uv E(T)d(u,v)=2
T T =
u vw
6
26VIT1984-2010
Creating Stars
Example 1
u v
a
b
c
x
y
z
Let T be a tree on ‘q’ edges
Counting all subtrees of the form of T
uv E(T)
d(v) 1 d(u) 1 = (d(u) 1) (d(v) 1)
2 2
(1,2) (2,1)N T T
7
26VIT1984-2010
Creating Stars
Example 2
Let T be a tree on ‘q’ edges
Counting all subtrees of the form of T
uv E(T)
d(v) 1 d(u) 1 = (d(u) 1) (d(v) 1)
3 3
(1,3) (3,1)N T T
8
u v
a
b
c
x
y
z
26VIT1984-2010
Creating Stars
9
v2
v V(T)
d(v)1. N(P )
2
2
uv E(T)
12. N(P ) d(u) 1 d(v) 1
2
2 (0,1) (1,0)
1N(P ) N T T
2
u v 1v
u v1u
26VIT1984-2010
Creating Stars
(1,0) (0,1)
uv E(T)
N T T d(u) 1 d(v) 1
10
26VIT1984-2010
Creating Stars
3
uv E(T)
3. N(P ) d(u) 1 d(v) 1
(1,1)
uv E(T)
N T d(u) 1 d(v) 1
3 (1,1)N(P ) N T
3
uv E(T)d(u,v) 2
14. N(P ) d(u) 1 d(v) 1
2
u w v1u
1vu w v
u v1v1u
(1,0,0) (0,0,1)
uv E(T)d(u,v) 2
N T T d(u) 1 d(v) 1
2 (1,0) (0,1)
1N(P ) N T T
2
11
26VIT1984-2010
Creating Stars
4
uv E(T)d(u,v) 2
5. N(P ) d(u) 1 d(v) 1
(1,0,1)
uv E(T)d(u,v) 2
N T d(u) 1 d(v) 1
4 (1,0,1)N(P ) N T
1vu w v1u
m 2
uv E(T)d(u,v) m
6. N(P ) d(u) 1 d(v) 1
u v
(m 1) vertices
1,3
v V(T)
d(v)7. N(K )
3
v
12
26VIT1984-2010
Creating Stars
1,3
uv E(T)
d(u) 1 d(v) 118. N(K )
2 23
1,3
uv E(T)d(u,v) 2
19. N(K ) d(w) 2
3
(0,1,0)
uv E(T)d(u,v) 2
N T d(w) 2
1,3 (0,1,0)
1N(K ) N T
3
u vw
1w
u
v
1v 2v
u
v
2u1u
(2,0) (0,2)
uv E(T)
d(u) 1 d(v) 1N T T
2 2
1,3 (2,0) (0,2)
1N(K ) N T T
3
13
26VIT1984-2010
Creating Stars
1,4
v V(T)
d(v)10. N(K )
4
v
1,4
uv E(T)
d(u) 1 d(v) 1111. N(K )
3 34
u
v
1v2
v 3v
u
v
1u 2
u3u
(3,0) (0,3)
uv E(T)
d(u) 1 d(v) 1N T T
3 3
1,4 (3,0) (0,3)
1N(K ) N T T
4
14
26VIT1984-2010
Creating Stars
1,4
uv E(T)d(u,v) 2
d(w) 2112. N(K )
26
(0,2,0)
uv E(T)d(u,v) 2
d(w) 2N T
2
1,4 (0,2,0)
1N(K ) N T
6
u vw
1w2w
v1,5
v V(T)
d(v)13. N(K )
5
15
26VIT1984-2010
Creating Stars
1,5
uv E(T)
(4,0) (0,4)
uv E(T)
1,5 (4,0) (0,4)
d(u) 1 d(v) 1114. N(K )
4 45
d(u) 1 d(v) 1 N T T
4 4
1 N(K ) N T T
5
1,5
uv E(T)d(u,v) 2
(0,3,0)
uv E(T)d(u,v) 2
1,5 (0,3,0)
d(w) 2115. N(K )
310
d(w) 2 N T
3
1 N(K ) N T
10
u vw
1w2w
3w
u
v
1u4u
3u2u
u
v
1v
2v4v
3v
16
26VIT1984-2010
Creating Stars
In General
1,n
v V(T)
d(v)N(K )
n
1,n
uv E(T)
d(u) 1 d(v) 11N(K )
n 1 n 1N
1,n
uv E(T)d(u,v) 2
d(w) 21N(K )
n n 2
2
(where ) n 2
17
Proposition 1: Let T be a tree with ‘q’ edges
2
2
v V(T) uv E(T)
i. d (v) d(u) d(v) 2N(P ) 2q
3 2 2
1,3 2
v V(T) uv E(T)
ii. d (v) d (u) d (v) 6N(K ) 6N(P ) 2q
4 3 3
1,4 1,3
v V(T) uv E(T)
2
iii. d (v) d (u) d (v) 24N(K ) 36N(K )
14N(P ) 2q
5 4 4
1,5 1,4
v V(T) uv E(T)
1,3 2
iv. d (v) d (u) d (v) 120N(K ) 240N(K )
150N(K ) 30N(P ) 2q
26VIT1984-2010
Creating Stars
Proof of Proposition 1
2
2
v V(T) v V(T)
d(v) d (v) d(v)i. N(P )
2 2
2
2
v V(T) v V(T)
d (v) 2N(P ) d(v)
2
2
v V(T) uv E(T)
d (v) d(u) d(v) 2N(P ) 2q
26VIT1984-2010
Creating Stars
1,3
v V(T) v V(T)
d(v) d(v)(d(v) 1)(d(v) 2)ii. N(K )
3 6
3 2
1,3
v V(T) v V(T) v V(T)
d (v) 6N(K ) 3 d (v) 2 d(v)
3 2 2
1,3 2
v V(T) uv E(T)
d (v) d (u) d (v) 6N(K ) 6N(P ) 2q
26VIT1984-2010
Creating Stars
1,4
v V(T)
v V(T)
d(v)iii. N(K )
4
d(v)(d(v) 1)(d(v) 2)(d(v) 3)
24
4 3 2
1,4
v V(T) v V(T) v V(T) v V(T)
d (v) 24N(K ) 6 d (v) 11 d (v) 6 d(v)
4 3 3
1,4 1,3
v V(T) uv E(T)
2
d (v) d (u) d (v) 24N(K ) 36N(K )
14N(P ) 2q
26VIT1984-2010
Creating Stars
1,5
v V(T)
v V(T)
d(v)iv. N(K )
5
d(v)(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)
120
5 4 3
1,5
v V(T) v V(T) v V(T)
2
v V(T) v V(T)
d (v) 120N(K ) 10 d (v) 35 d (v)
50 d (v) 24 d(v)
5 4 4
1,5 1,4
v V(T) uv E(T)
1,3 2
d (v) d (u) d (v) 120N(K ) 240N(K )
150N(K ) 30N(P ) 2q
26VIT1984-2010
Creating Stars
Verification of Proposition 1 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
2**xy E( )
d(x) d(y) 2N (P ) 2(2q)
T
T
L.H.S
2
* uv E( )xy E( )
d(x) d(y) (d(u) 2) (d(v) 2) 2N (P ) 6q
TTT
2
uv E( )
i. d(u) d(v) 2N (P ) 2q
TT
2 2*2N (P ) 2(2q) 2N (P ) 6q
TT
R.H.S
26VIT1984-2010
Creating Stars
2 2
1,3 2
uv E( )
ii. d (u) d (v) 6N (K ) 6N (P ) 2q
T TT
2 2
1,3 2
xy E( )
d (x) d (y) 6N (K ) 6N (P ) 2(2q)
* *T T*
T
2 2 2 2
1,3 2
uv E( )xy E( )
d (x) d (y) (d (u) 4) (d (v) 4)
6N (K ) 6N (P ) 10q
* TT
T T
L.H.S
R.H.S
1,3 26N (K ) 6N (P ) 2(2q) T T* *
1,3 26N (K ) 6N (P ) 10q T T
1,3 1,3N (K ) N (K )* T
T
26VIT1984-2010
Creating Stars
3 3
1,4 1,3 2
uv E( )
iii. d (u) d (v) 24N (K ) 36N (K ) 14N (P ) 2q
T T TT
3 3
1,4 1,3 2
xy E( )
d (x) d (y) 24N (K ) 36N (K ) 14N (P ) 2(2q)
* * *T T T*
T
3 3 3 3
1,4 1,3 2
uv E( )xy E( )
d (x) d (y) (d (u) 8) (d (v) 8)
24N (K ) 36N (K ) 14N (P ) 18q
* TT
T T T
L.H.S
R.H.S
1,4 1,3 224N (K ) 36N (K ) 14N (P ) 2(2q) * * *
T T T 1,4 1,4* TTN (K ) N (K )
1,4 1,3 224N (K ) 36N (K ) 14N (P ) 18q T T T
26VIT1984-2010
Creating Stars
4 4
1,5 1,4
1,3 2
uv E( )
iv. d (u) d (v) 120N (K ) 240N (K )
150N (K ) 30N (P ) 2q
T TT
T T
4 4
1,5 1,4 1,3
2
xy E( )
d (x) d (y) 120N (K ) 240N (K ) 150N (K )
30N (P ) 2(2q)
* * *T T T*
T
*T
4 4 4 4
uv E( )xy E( )
d (x) d (y) (d (u) 16) (d (v) 16)
* TT
1,5 1,4 1,3 2120N (K ) 240N (K ) 150N (K ) 30N (P ) 34q T T T T
1,5 1,4 1,3 2120N (K ) 240N (K ) 150N (K ) 30N (P ) 2(2q) * * * *
T T T T
1,5 1,4 1,3 2120N (K ) 240N (K ) 150N (K ) 30N (P ) 34q T T T T
1,5 1,5N (K ) N (K )* T
T
L.H.S
R.H.S
26
26VIT1984-2010
Creating Stars
Proposition 2: Let T be a tree with ‘q’ edges.
1,3 2
uv E(T)d(u,v) 2
i. d(w) 3N(K ) 2N(P )
4
1,6 1,5 1,4
uv E(T)d(u,v) 2
1,3 2
iv. d (w) 360N(K ) 840N(K ) 660N(K )
195N(K ) 16N(P )
3
1,5 1,4 1,3 2
uv E(T)d(u,v) 2
iii. d (w) 60N(K ) 108N(K ) 57N(K ) 8N(P )
2
1,4 1,3 2
uv E(T)d(u,v) 2
ii. d (w) 12N(K ) 15N(K ) 4N(P )
uw,wv E(T)
26VIT1984-2010
Creating Stars
1,3
uv E(T)d(u,v) 2
1i. N(K ) (d(w) 2)
3
Proof of Proposition 2
1,3 2
uv E(T)d(u,v) 2
d(w) 3N(K ) 2N(P )
1,4
uv E(T)d(u,v) 2
d(w) 21ii. N(K )
26
2
1,4
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
12N(K ) d(w) 2 d(w) 3 d (w) 5d(w) 6
2
1,4 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d (w) 12N(K ) 5 d(w) 6N(P )
2
1,4 1,3 2
uv E(T)d(u,v) 2
d (w) 12N(K ) 15N(K ) 4N(P )
26VIT1984-2010
Creating Stars
1,5
uv E(T)d(u,v) 2
d(w) 21iii. N(K )
310
1,5
uv E(T)d(u,v) 2
60N(K ) (d(w) 2)(d(w) 3)(d(w) 4)
3 2
1,5
uv E(T)d(u,v) 2
60N(K ) d (w) 9d (w) 26d(w) 24
3 2
1,5 2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
d (w) 60N(K ) 9 d (w) 26 d(w) 24N(P )
3
1,5 1,4 1,3 2
uv E(T)d(u,v) 2
d (w) 60N(K ) 108N(K ) 57N(K ) 8N(P )
26VIT1984-2010
Creating Stars
1,6
uv E(T)d(u,v) 2
d(w) 21iv. N(K )
415
1,6
uv E(T)d(u,v) 2
360N(K ) (d(w) 2)(d(w) 3)(d(w) 4)(d(w) 5)
4 3 2
1,6
uv E(T)d(u,v) 2
360N(K ) d (w) 14d (w) 71d (w) 154d(w) 120
4 3 2
1,6 2
uv E(T) uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2 d(u,v) 2
d (w) 360N(K ) 14 d (w) 71 d (w) 154 d(w) 120N(P )
4
1,6 1,5 1,4
uv E(T)d(u,v) 2
1,3 2
d (w) 360N(K ) 840N(K ) 660N(K )
195N(K ) 16N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 2 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
1,3 2
uv E( )d(u,v) 2
i. d(w) 3N (K ) 2N (P )
T TT
1,3 2
xy E( )d(x,y) 2
d(z) 3N (K ) 2N (P )
* *T T*
T
3 2
uv E( ) v V( )xy E( )d(x,y) 2
v V( ) v V( )
d(v)d(z) (2) d(v)
2
1 1 d (v) d (v) 2q
2 2
* T TT
T T
L.H.S
26VIT1984-2010
Creating Stars
1,3 23N (K ) 2N (P ) 2q T T
1,3 2 1,3 23N (K ) 2N (P ) 3N (K ) 2N (P ) 2q * * T T
T T
R.H.S
2
1,4 1,3 2
uv E( )d(u,v) 2
ii. d (w) 12N (K ) 15N (K ) 4N (P )
T T T
T
2
1,4 1,3 2
xy E( )d(x,y) 2
d (z) 12N (K ) 15N (K ) 4N (P )
* * *T T T*T
26VIT1984-2010
Creating Stars
2 2
4 3
uv E( ) v V( )xy E( )d(x,y) 2
v V( ) v V( )
d(v)d (z) (4) d (v)
2
1 1 d (v) d (v) 4q
2 2
* T TT
T T
1,3 1,3 212N (K ) 15N (K ) 4N (P ) 4q T T T
1,4 1,3 2
1,4 1,3 2
12N (K ) 15N (K ) 4N (P )
12N (K ) 15N (K ) 4N (P ) 4q
* * *T T T
T T T
L.H.S
R.H.S
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Creating Stars
3
1,5 1,4 1,3 2
uv E( )d(u,v) 2
iii. d (w) 60N (K ) 108N (K ) 57N (K ) 8N (P )
T T T TT
3
1,5 1,4 1,3 2
xy E( )d(x,y) 2
d (z) 60N (K ) 108N (K ) 57N (K ) 8N (P )
* * * *T T T T*
T
3 3
5 4
uv E( ) v V( )xy E( )d(x,y) 2
v V( ) v V( )
d(v)d (z) (8) d (v)
2
1 1 d (v) d (v) 8q
2 2
* T TT
T T
L.H.S
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Creating Stars
4
1,6 1,5 1,4
1,3 2
uv E( )d(u,v) 2
iv. d (w) 360N (K ) 840N (K ) 660N (K )
195N (K ) 16N (P )
T T TT
T T
4
1,6 1,5 1,4
1,3 2
xy E( )d(x,y) 2
d (z) 360N (K ) 840N (K ) 660N (K )
195N (K ) 16N (P )
* * *T T T*
T
* *T T
R.H.S
1,5 1,4 1,3 2
1,5 1,4 1,3 2
60N (K ) 108N (K ) 57N (K ) 8N (P )
60N (K ) 108N (K ) 57N (K ) 8N (P ) 8q
* * * *T T T T
T T T T
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Creating Stars
4 4
6 5
uv E( ) v V( )xy E( )d(x,y) 2
v V( ) v V( )
d(v)d (z) (16) d (v)
2
1 1 d (v) d (v) 16q
2 2
* T TT
T T
1,6 1,5 1,4
1,3 2
360N (K ) 840N (K ) 660N (K )
195N (K ) 16N (P ) 16q
T T T
T T
L.H.S
R.H.S
1,6 1,5 1,4 1,3 2360N (K ) 840N (K ) 660N (K ) 195N (K ) 16N (P ) * * * * *
T T T T T
1,6 1,5 1,4
1,3 2
360N (K ) 840N (K ) 660N (K )
195N (K ) 16N (P ) 16q
T T T
T T
36
26VIT1984-2010
Creating Stars
uv E(T)
17. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
3 3
uv E(T)
16. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
2 2
uv E(T)
18. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
4 4
37
26VIT1984-2010
Creating Stars
uv E(T)
d(u) 1 d(v) 119. N
2 2
uv E(T)
20. N
d(u) 1 d(v) 1 d(u) 1 d(v) 1
2 3 3 2
uv E(T)
d(u) 1 d(v) 121. N =
3 3
uv E(T)
22. N
d(u) 1 d(v) 1 d(u) 1 d(v) 1
2 4 4 2
38
26VIT1984-2010
Creating Stars
3 2
uv E(T)
i. d(u)d(v) N(P ) 2N(P ) q
Proposition 3: Let T be a tree with ‘q’ edges
1,3 3
uv E(T)
2
ii. d(u)d(v) d(u) d(v) 2N 6N(K ) 6N(P )
8N(P ) 2q
39
26VIT1984-2010
Creating Stars
2 2
uv E(T)
1,4 1,3
iii. d(u)d(v) d (u) d (v) 6N 12N
24N(K ) 36N(K )
3 214N(P ) 16N(P ) 2q
3 3
uv E(T)
iv. d(u)d(v) d (u) d (v) 24N 60N
50N
1,5 1,4
1,3 3 2
120N(K ) 240N(K )
150N(K ) 30N(P ) 32N(P ) 2q
Proof of Proposition 3
3
uv E(T)
i. N(P ) d(u) 1 d(v) 1
3
uv E(T) uv E(T)
d(u)d(v) N(P ) d(u) d(v) q
2 2
uv E(T)
d(u)d(v) d(u) d(v) d (u) d (v)2N
5 d(u) d(v) 6d(u)d(v) 4
uv E(T)
d(v) 1 d(u) 1ii. N = (d(u) 1) (d(v) 1)
2 2
3 2
uv E(T)
d(u)d(v) N(P ) 2N(P ) q
2 2
uv E(T) uv E(T)
uv E(T) uv E(T)
d(u)d(v) d(u) d(v) 2N d (u) d (v)
5 d(u) d(v) 6 d(u)d(v) 4q
1,3 3
uv E(T)
2
ii. d(u)d(v) d(u) d(v) 2N 6N(K ) 6N(P )
8N(P ) 2q
uv E(T)
d(v) 1 d(u) 1iii. N = (d(u) 1) (d(v) 1)
3 3
uv E(T)
(d(u) 1)(d(v) 1)(d(v) 2)(d(v) 3)6N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(v) 1)
2 2 3 3
2 2
uv E(T)
6 d (u) d (v) d (u) d (v) 17 d(u) d(v)
6N = d(u)d(v) d (u) d (v) 6d(u)d(v) d(u) d(v)
22d(u)d(v) 12
2 2 3 3
uv E(T) uv E(T)
2 2
uv E(T) uv E(T)
d(u)d(v) d (u) d (v) 6N d (u) d (v) 12q
6 d (u) d (v) 17 d(u) d(v)
uv E(T) uv E(T)
6 d(u)d(v) d(u) d(v) 22 d(u)d(v)
3 3
uv E(T)
iv. d(u)d(v) d (u) d (v) 24N 60N
50N
1,5 1,4
1,3 3 2
120N(K ) 240N(K )
150N(K ) 30N(P ) 32N(P ) 2q
26VIT1984-2010
Creating Stars
uv E(T)
d(v) 1 d(u) 1iv. N = (d(u) 1) (d(v) 1)
4 4
uv E(T)
(d(u) 1)(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)24N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(u) 4)(d(v) 1)
3 3
4 4
3 3 2 2uv E
2 2
d(u)d(v) d (u) d (v) 100d(u)d(v) 48
35d(u)d(v) d(u) d(v) d (u) d (v)24N =
10 d (u) d (v) 35 d (u) d (v)
74 d(u) d(v) 10d(u)d(v) d (u) d (v)
(T)
26VIT1984-2010
Creating Stars
3 3
uv E(T) uv E(T)
2 2
uv E(T) uv E(T)
d(u)d(v) d (u) d (v) 24N 35 d(u)d(v) d(u) d(v)
10 d(u)d(v) d (u) d (v) 100 d(u)d(v)
4 4 3 3
uv E(T) uv E(T)
2 2
uv E(T) uv E(T)
d (u) d (v) 10 d (u) d (v)
35 d (u) d (v) 74 d(u) d(v) 48q
3 3
uv E(T)
iv. d(u)d(v) d (u) d (v) 24N 60N
50N
1,5 1,4
1,3 3 2
120N(K ) 240N(K )
150N(K ) 30N(P ) 32N(P ) 2q
26VIT1984-2010
Creating Stars
Verification of Proposition 3 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
3 2
uv E( )
i. d(u)d(v) N (P ) 2N (P ) q
T TT
3 2
xy E( )
d(x)d(y) N (P ) 2N (P ) (2q)
* *T T*T
uv E( ) uv E( )xy E( )
d(x)d(y) d(u)(2) (2)d(v) 2 d(u) d(v)
* T TT
24N (P ) 4q T
3 2 2N (P ) 2N (P ) (2q) 4N (P ) 4q * * T
T T
L.H.S
R.H.S
26VIT1984-2010
Creating Stars
uv E( )xy E( )
d(x)d(y) d(x) d(y) d(u)(2) d(u) 2 (2)d(v) 2 d(v)
* TT
2 2
uv E( ) uv E( )
2 d (u) d (v) 4 d(u) d(v)
T T
1,3 212N (K ) 20N (P ) 12q T T
L.H.S
1,3 3
2
uv E( )
ii. d(u)d(v) d(u) d(v) 2N 6N (K ) 6N (P )
8N (P ) 2q
T T TT
T
1,3 3
2
xy E( )
d(x)d(y) d(x) d(y) 2N 6N (K ) 6N (P )
8N (P ) 2(2q)
* * *T T T*T
*T
26VIT1984-2010
Creating Stars
R.H.S
1,3N 3N (K )
* TT
2 2
uv E( )
1,4 1,3
iii. d(u)d(v) d (u) d (v) 6N 12N
24N (K ) 36N (K )
T T
T
T T 3 214N (P ) 16N (P ) 2q T T
*
2 2
xy E( )
1,4 1,3 3T
d(x)d(y) d (x) d (y) 6N 12N
24N (K ) 36N (K ) 14N (P ) 16N (P
* *
*
* * *
T TT
T T T 2 ) 2(2q)
1,3 212N (K ) 20N (P ) 12q T T
1,3 3 22N 6N (K ) 6N (P ) 8N (P ) 2(2q)
* * * *T T T T
26VIT1984-2010
Creating Stars
L.H.S
R.H.S
2 2 2 2
uv E( )xy E( )
d(x)d(y) d (x) d (y) d(u)(2) d (u) 4 (2)d(v) 4 d (v)
* TT
3 3
uv E( ) uv E( )
2 d (u) d (v) 8 d(u) d(v)
T T
1,4 1,3 248N (K ) 72N (K ) 44N (P ) 20q T T T
1,4 1,3
3 2
6N 12N 24N (K ) 36N (K )
14N (P ) 16N (P ) 2(2q)
* * * *
* *
T T T T
T T
1,4N 4N (K )
* TT
1,4 1,3 248N (K ) 72N (K ) 44N (P ) 20q T T T
26VIT1984-2010
Creating Stars
3 3
uv E( )
iv. d(u)d(v) d (u) d (v) 24N 60N
50N
T T
T
T 1,5 1,4
1,3 3 2
120N (K ) 240N (K )
150N (K ) 30N (P ) 32N (P ) 2q
T T
T T T
3 3
xy E( )
d(x)d(y) d (x) d (y) 24N 60N
50N
* *
*
*
T TT
T 1,5 1,4
1,3 3 2
120N (K ) 240N (K )
150N (K ) 30N (P ) 32N (P ) 2(2q)
* *
* * *
T T
T T T
3 3 3 3
uv E( )xy E( )
d(x)d(y) d (x) d (y) d(u)(2) d (u) 8 (2)d(v) 8 d (v)
* TT
L.H.S
26VIT1984-2010
Creating Stars
4 4
uv E( ) uv E( )
2 d (u) d (v) 16 d(u) d(v)
T T
1,5 1,4 1,3 2240N (K ) 480N (K ) 300N (K ) 92N (P ) 36q T T T T
1,5 1,4 1,3 3 2
24N 60N 50N
120N (K ) 240N (K ) 150N (K ) 30N (P ) 32N (P ) 2(2q)
* * *
* * * * *
T T T
T T T T T
R.H.S
1,5N 5N (K )
* TT
1,5 1,4 1,3 2240N (K ) 480N (K ) 300N (K ) 92N (P ) 36q T T T T
51
26VIT1984-2010
Creating Stars
Proposition 4: Let T be a tree with ‘q’ edges
2 2
1,3
uv E(T)
3 2
i. d (u)d (v) 4N 6N 6N(K )
9N(P ) 6N(P ) q
2 2
uv E(T)
ii. d (u)d (v) d(u) d(v) 12N 18N
48N
1,4 1,3 3 2
50N
24N(K ) 42N(K ) 42N(P ) 20N(P ) 2q
*T
52
26VIT1984-2010
Creating Stars
2 2 2 2
uv E(T)
iii. d (u)d (v) d (u) d (v) 48N 72N
180N
180N
200N 120N
1,5 1,4 1,3 3 2120N(K ) 240N(K ) 156N(K ) 90N(P ) 36N(P ) 2q
3 3
uv E(T)
iv. d (u)d (v) 36N 72N 42N
84N
1,4 1,3 3 2
144N
24N(K ) 36N(K ) 49N(P ) 14N(P ) q
26VIT1984-2010
Creating Stars
uv E(T)
d(u) 1 d(v) 1i. N =
2 2
Proof of Proposition 4
uv E(T)
4N (d(u) 1)(d(u) 2)(d(v) 1)(d(v) 2)
2 2
2 2
uv E(T)
d (u)d (v) 3d(u)d(v) d(u) d(v)
4N 2 d (u) d (v) 6 d(u) d(v)
9d(u)d(v) 4
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)
3 2
uv E(T) v V(T) v V(T)
d (u)d (v) 4N 3 d(u)d(v) d(u) d(v)
9 d(u)d(v) 2 d (v) 6 d (v) 4q
2 2
1,3
uv E(T)
3 2
i. d (u)d (v) 4N 6N 6N(K )
9N(P ) 6N(P ) q
26VIT1984-2010
Creating Stars
uv E(T)
d(u) 1 d(v) 1 d(u) 1 d(v) 1ii. N =
2 3 3 2
uv E(T)
(d(u) 1)(d(u) 2)(d(v) 1)(d(v) 2)(d(v) 3)12N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(v) 1)(d(v) 2)
2 2
2 2 3 3
2 2uv E
2 2
d (u)d (v) d(u) d(v) 40 d(u) d(v)
18 d (u) d (v) 2 d (u) d (v)12N =
29d(u)d(v) d(u) d(v) 12d (u)d (v)
3d(u)d(v) d (u) d (v) 66d(u)d(v) 24
(T)
26VIT1984-2010
Creating Stars
2 2 4 3
uv E(T) v V(T) v V(T)
2 2 2
v V(T) uv E(T)
d (u)d (v) d(u) d(v) 12N 2 d (v) 18 d (v)
40 d (v) 3 d(u)d(v) d (u) d (v)
uv E(T) uv E(T)
2 2
uv E(T)
29 d(u)d(v) d(u) d(v) 66 d(u)d(v)
12 d (u)d (v) 24q
2 2
uv E(T)
ii. d (u)d (v) d(u) d(v) 12N 18N
48N
1,4 1,3 3 2
50N
24N(K ) 42N(K ) 42N(P ) 20N(P ) 2q
uv E(T)
d(u) 1 d(v) 1 d(u) 1 d(v) 1iii. N =
2 4 4 2
2 2 2 2 4 4
3 3 3 3
2 2 2 2
2 2
d (u)d (v) d (u) d (v) 2 d (u) d (v)
3d(u)d(v) d (u) d (v) 20 d (u) d (v)
10d (u)d (v) d(u) d(v) 94 d (u) d (v)48N =
30d(u)d(v) d (u) d (v) 172 d(u) d(
uv E(T)
2 2
v)
155d(u)d(v) d(u) d(v) 300d(u)d(v)
70d (u)d (v) 96
uv E(T)
(d(u) 1)(d(u) 2)
(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)48N =
(d(u) 1)(d(u) 2)
(d(u) 3)(d(u) 4)(d(v) 1)(d(v) 2)
2 2 2 2 5
uv E(T) v V(T)
3 3 4
uv E(T) v V(T)
2
d (u)d (v) d (u) d (v) 48N 2 d (v)
3 d(u)d(v) d (u) d (v) 20 d (v)
10 d (u)d
2 3
uv E(T) v V(T)
2 2 2
uv E(T) v V(T)
2 2
uv E(T) uv E(T)
(v) d(u) d(v) 94 d (v)
30 d(u)d(v) d (u) d (v) 172 d (v)
300 d(u)d(v) 70 d (u)d (v)
v V(T)
155 d(u)d(v) d(u) d(v) 96q
2 2 2 2
uv E(T)
iii. d (u)d (v) d (u) d (v) 48N 72N
180N
180N
200N 120N
1,5 1,4 1,3 3 2120N(K ) 240N(K ) 156N(K ) 90N(P ) 36N(P ) 2q
26VIT1984-2010
Creating Stars
uv E(T)
d(u) 1 d(v) 1iv. N =
3 3
uv E(T)
(d(u) 1)(d(u) 2)(d(u) 3)36N =
(d(v) 1)(d(v) 2)(d(v) 3)
3 3 2 2
2 2 3 3
2 2
2 2
d (u)d (v) 36d (u)d (v) 121d(u)d(v)
11d(u)d(v) d (u) d (v) 6 d (u) d (v)36N =
66d(u)d(v) d(u) d(v) 36 d (u) d (v)
6d (u)d (v) d(u) d(v) 66 d(u) d(v) 36
uv E(T)
26VIT1984-2010
Creating Stars
3 3 2 2
uv E(T) uv E(T) uv E(T)
4 3
uv E(T) v V(T) v V(T)
d (u)d (v) 36N 36 d (u)d (v) 121 d(u)d(v)
66 d(u)d(v) d(u) d(v) 6 d (v) 36 d (v)
2 2 2
v V(T) uv E(T)
2 2
uv E(T)
66 d (v) 11 d(u)d(v) d (u) d (v)
6 d (u)d (v) d(u) d(v) 36q
3 3
uv E(T)
iv. d (u)d (v) 36N 72N 42N
84N
1,4 1,3 3 2
144N
24N(K ) 36N(K ) 49N(P ) 14N(P ) q
26VIT1984-2010
Creating Stars
Verification of Proposition 4 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
2 2
1,3
uv E( )
3 2
i. d (u)d (v) 4N 6N 6N (K )
9N (P ) 6N (P ) q
T T T
T
T T
2 2
1,3
xy E( )
3 2
d (x)d (y) 4N 6N 6N (K )
9N (P ) 6N (P ) (2q)
* * *
*
* *
T T TT
T T
2 2 2 2 2 2
uv E( ) uv E( )xy E( )
d (x)d (y) d (u)(4) (4)d (v) 4 d (u) d (v)
* T TT
L.H.S
1,3 224N (K ) 24N (P ) 8q T T
26VIT1984-2010
Creating Stars
R.H.S
1,3 3
2
4N 6N 6N (K ) 9N (P )
6N (P ) (2q)
* * * *
*
T T T T
T
N 0
*T
1,3 224N (K ) 24N (P ) 8q T T
2 2
uv E( )
ii. d (u)d (v) d(u) d(v) 12N 18N
48N
T T
T
T
1,4 1,3 3 2
50N
24N (K ) 42N (K ) 42N (P ) 20N (P ) 2q
T
T T T T
26VIT1984-2010
Creating Stars
2 2
xy E( )
d (x)d (y) d(x) d(y) 12N 18N
48N
* *
*
*
T TT
T
1,4 1,3 3 2
50N
24N (K ) 42N (K ) 42N (P ) 20N (P ) 2(2q)
*
* * * *
T
T T T T
2 2 2 2
uv E( )xy E( )
d (x)d (y) d(x) d(y) d (u)(4) d(u) 2 (4)d (v) 2 d(v)
* TT
L.H.S
3 3 2 2
uv E( ) uv E( )
4 d (u) d (v) 8 d (u) d (v)
T T
1,4 1,3 296N (K ) 192N (K ) 104N (P ) 24q T T T
26VIT1984-2010
Creating Stars
R.H.S
1,4 1,3
12N 18N 48N
50N 24N (K ) 42N (K ) 42N
* * *
* * *
T T T
T T T T 3 2(P ) 20N (P ) 2(2q) * *T
*TN 0
1,4 1,3 296N (K ) 192N (K ) 104N (P ) 24q T T T
26VIT1984-2010
Creating Stars
2 2 2 2
uv E( )
iii. d (u)d (v) d (u) d (v) 48N 72N
180N
T T
T
T 180N
200N 120N
T
T T
1,5 1,4 1,3 3 2120N (K ) 240N (K ) 156N (K ) 90N (P ) 36N (P ) 2q
T T T T T
2 2 2 2
xy E( )
d (x)d (y) d (x) d (y) 48N 72N
180N
* *
*
*
T TT
T180N
200N 120N
*
* *
T
T T
1,5 1,4 1,3 3 2120N (K ) 240N (K ) 156N (K ) 90N (P ) 36N (P ) 2(2q)
* * * * *T T T T T
26VIT1984-2010
Creating Stars
2 2 2 2
xy E( )
2 2 2 2
uv E( )
d (x)d (y) d (x) d (y)
d (u)(4) d (u) 4 (4)d (v) 4 d (v)
*T
T
L.H.S
4 4 2 2
uv E( ) uv E( )
4 d (u) d (v) 16 d (u) d (v)
T T
1,5 1,4 1,3 2480N (K ) 960N (K ) 696N (K ) 216N (P ) 40q T T T T
R.H.S
48N 72N 180N
180N 200N
* * *
* *
T T T
T T
1,5 1,4 1,3 3 2
120N
120N (K ) 240N (K ) 156N (K ) 90N (P ) 36N (P ) 2(2q)
*
* * * * *
T
T T T T T
1,5 1,4 1,3 2480N (K ) 960N (K ) 696N (K ) 216N (P ) 40q T T T T
26VIT1984-2010
Creating Stars
3 3
uv E( )
iv. d (u)d (v) 36N 72N 42N
84 N
T T T
T
T
1,4 1,3 3 2
144N
24N (K ) 36N (K ) 49N (P ) 14N (P ) q
T
T T T T
3 3
xy E( )
d (x)d (y) 36N 72N 42N
84N
* * *
*
*
T T TT
T
1,4 1,3 3 2
144N
24N (K ) 36N (K ) 49N (P ) 14N (P ) (2q)
*
* * * *
T
T T T T
3 3 3 3 3 3
uv E( ) uv E( )xy E( )
d (x)d (y) d (u)(8) (8)d (v) 8 d (u) d (v)
* T TT
L.H.S
1,4 1,3 2192N (K ) 288N (K ) 112N (P ) 16q T T T
26VIT1984-2010
Creating Stars
R.H.S
36N 72N 42N
84N 144N
* * *
* *
T T T
T T
1,4 1,3 3 224N (K ) 36N (K ) 49N (P ) 14N (P ) (2q)
* * * *T T T T
N 0
*T
1,4 1,3 2192N (K ) 288N (K ) 112N (P ) 16q T T T
69
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 123. N
2 2
uv E(T)d(u,v) 2
d(u) 1 d(v) 124. N
3 3
uv E(T)d(u,v) 2
d(u) 1 d(v) 125. N
4 4
uv E(T)d(u ,v) 2
26. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
2 2
70
26VIT1984-2010
Creating Stars
uv E(T)d(u ,v) 2
27. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
3 3
uv E(T)d(u ,v) 2
28. N
d(v) 1 d(u) 1(d(u) 1) (d(v) 1)
4 4
uv E(T)d(u,v) 2
d(u) 1 d(v) 129. N
2 2
71
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1 d(u) 1 d(v) 130. N
2 3 3 2
uv E(T)d(u,v) 2
d(u) 1 d(v) 1 d(u) 1 d(v) 131. N
2 4 4 2
uv E(T)d(u,v) 2
d(u) 1 d(v) 132. N
3 3
72
26VIT1984-2010
Creating Stars
Proposition 5: Let T be a tree with ‘q’ edges
3 2
uv E(T)d(u,v) 2
i. d(u) d(v) 2N(P ) 2N(P )
2 2
3 2
uv E(T)d(u,v) 2
ii. d (u) d (v) 2N 6N(P ) 2N(P )
3 3
uv E(T)d(u,v) 2
3 2
iii. d (u) d (v) 6N 12N
14N(P ) 2N(P )
4 4
uv E(T)d(u,v) 2
3
iv. d (u) d (v) 24N 60N
50N 30N(P
2) 2N(P )
26VIT1984-2010
Creating Stars
3
uv E(T)d(u,v) 2
1i. N(P ) (d(u) 1) (d(v) 1)
2
3 2
uv E(T)d(u,v) 2
d(u) d(v) 2N(P ) 2N(P )
uv E(T)d(u,v) 2
d(u) 1 d(v) 1ii. N
2 2
uv E(T)d(u,v) 2
2N (d(u) 1)(d(u) 2) (d(v) 1)(d(v) 2)
Proof of Proposition 5
26VIT1984-2010
Creating Stars
2 2
uv E(T)d(u,v) 2
2N d (u) d (v) 3 d(u) d(v) 4
2 2
2
uv E(T) {u,v} E(T)d(u,v) 2 d(u,v) 2
d (u) d (v) 2N 3 d(u) d(v) 4N(P )
2 2
3 2
uv E(T)d(u,v) 2
ii. d (u) d (v) 2N 6N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1iii. N
3 3
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)(d(u) 3)6N
(d(v) 1)(d(v) 2)(d(v) 3)
3 3 2 2
uv E(T)d(u,v) 2
6N d (u) d (v) 6 d (u) d (v) 11 d(u) d(v) 12
26VIT1984-2010
Creating Stars
3 3 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2
uv E(T)d(u,v) 2
d (u) d (v) 6N 6 d (u) d (v)
11 d(u) d(v) 12N(P )
3 3
uv E(T)d(u,v) 2
3 2
iii. d (u) d (v) 6N 12N
14N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1iv. N
4 4
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)(d(u) 3)(d(u) 4)24N
(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)
4 4 3 3
2 2uv E(T)d(u,v) 2
d (u) d (v) 10 d (u) d (v)24N
35 d (u) d (v) 50 d(u) d(v) 48
26VIT1984-2010
Creating Stars
4 4 3 3
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d (u) d (v) 24N 10 d (u) d (v)
35 d (u) d (v) 50 d(u) d(v) 48N
2(P )
4 4
uv E(T)d(u,v) 2
3
iv. d (u) d (v) 24N 60N
50N 30N(P
2) 2N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 5 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
3 2
uv E( )d(u,v) 2
i. d(u) d(v) 2N (P ) 2N (P )
T TT
3 2
xy E( )d(x,y) 2
d(x) d(y) 2N (P ) 2N (P )
* *T T*T
uv E( ) v V( )xy E( )
d(x,y) 2
d(v)d(x) d(y) d(u) d(v) 4
2
* T TT
L.H.S
R.H.S
26N (P ) 2q T
26VIT1984-2010
Creating Stars
R.H.S
3 2 22N (P ) 2N (P ) 6N (P ) 2q * * T
T T
2 2
3 2
uv E( )d(u,v) 2
ii. d (u) d (v) 2N 6N (P ) 2N (P )
T T T
T
2 2
3 2
xy E( )d(x,y) 2
d (x) d (y) 2N 6N (P ) 2N (P )
* * *
T T T*T
2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x) d (y) d (u) d (v) 8
2
* T TT
L.H.S
1,3 26N (K ) 14N (P ) 2q T T
26VIT1984-2010
Creating Stars
R.H.S
3 22N 6N (P ) 2N (P )
* * *T T T
1,3 26N (K ) 14N (P ) 2q T T
3 3
3 2
uv E( )d(u,v) 2
iii. d (u) d (v) 6N 12N
14N (P ) 2N (P )
T TT
T T
3 3
3 2
xy E( )d(x,y) 2
d (x) d (y) 6N 12N
14N (P ) 2N (P )
* *T T*T
* *T T
26VIT1984-2010
Creating Stars
3 3 3 3
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x) d (y) d (u) d (v) 16
2
* T TT
L.H.S
R.H.S
1,4 1,3 224N (K ) 36N (K ) 30N (P ) 2q T T T
3 26N 12N 14N (P ) 2N (P )
* * * *T T T T
1,4 1,3 224N (K ) 36N (K ) 30N (P ) 2q T T T
26VIT1984-2010
Creating Stars
4 4
uv E( )d(u,v) 2
iv. d (u) d (v) 24N 60N
50N 3
T TT
T 3 20N (P ) 2N (P )T T
4 4
3
xy E( )d(x,y) 2
d (x) d (y) 24N 60N
50N 30N (P ) 2
* *T T*T
* *T T
2N (P )*
T
26VIT1984-2010
Creating Stars
R.H.S
3 2
24N 60N 50N
30N (P ) 2N (P )
* * *T T T
* *T T
1,5 1,4 1,3 2120N (K ) 240N (K ) 150N (K ) 62N (P ) 2q T T T T
4 4 4 4
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x) d (y) d (u) d (v) 32
2
* T TT
L.H.S
1,5 1,4 1,3 2120N (K ) 240N (K ) 150N (K ) 62N (P ) 2q T T T T
85
26VIT1984-2010
Creating Stars
Proposition 6: Let T be a tree with ‘q’ edges
4 3 2
uv E(T)d(u,v) 2
i. d(u)d(v) N(P ) 2N(P ) N(P )
uv E(T)d(u ,v) 2
4 3 2
ii. d(u)d(v) d(u) d(v) 2N 2N
6N(P ) 8N(P ) 2N(P )
86
26VIT1984-2010
Creating Stars
2 2
uv E(T)d(u,v) 2
iii. d(u)d(v) d (u) d (v) 6N 6N
12N
4 3 2
12N
14N(P ) 16N(P ) 2N(P )
3 3
uv E(T)d(u,v) 2
iv. d(u)d(v) d (u) d (v) 24N 24N
60N
60N
50N 50N
4 3 230N(P ) 32N(P ) 2N(P )
26VIT1984-2010
Creating Stars
Proof of Proposition 6
4
uv E(T)d(u,v) 2
i. N(P ) d(u) 1 d(v) 1
4 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d(u)d(v) N(P ) d(u) d(v) N(P )
4 3 2
uv E(T)d(u,v) 2
d(u)d(v) N(P ) 2N(P ) N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(v) 1 d(u) 1ii. N (d(u) 1) (d(v) 1)
2 2
uv E(T)d(u,v) 2
(d(u) 1)(d(v) 1)(d(v) 2)2N
(d(u) 1)(d(u) 2)(d(v) 1)
2 2
uv E(T)d(u,v) 2
d(u)d(v) d(u) d(v) d (u) d (v)2N
5 d(u) d(v) 6d(u)d(v) 4
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,
d(u)d(v) d(u) d(v) 2N d (u) d (v)
5 d(u) d(v) 6 d(u)d(v)
2
v) 2
4N(P )
uv E(T)d(u ,v) 2
4 3 2
ii. d(u)d(v) d(u) d(v) 2N 2N
6N(P ) 8N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(v) 1 d(u) 1iii. N = (d(u) 1) (d(v) 1)
3 3
uv E(T)d(u,v) 2
(d(u) 1)(d(v) 1)(d(v) 2)(d(v) 3)6N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(v) 1)
2 2 3 3
2 2
uv E(T)d(u,v) 2
6 d (u) d (v) d (u) d (v) 17 d(u) d(v)
6N = d(u)d(v) d (u) d (v) 6d(u)d(v) d(u) d(v)
22d(u)d(v) 12
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
3 3 2 2
uv E(T)d(u,v) 2
d(u)d(v) d (u) d (v) 6N 22 d(u)d(v)
d (u) d (v) 6 d (u) d (v)
uv E(T)d(u,v) 2
uv E(T)d(u,v) 2
2
uv E(T)d(u,v) 2
6 d(u)d(v) d(u) d(v)
17 d(u) d(v) 12N(P )
2 2
uv E(T)d(u,v) 2
iii. d(u)d(v) d (u) d (v) 6N 6N
12N
4 3 2
12N
14N(P ) 16N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(v) 1 d(u) 1iv. N = (d(u) 1) (d(v) 1)
4 4
uv E(T)d(u,v) 2
(d(u) 1)(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)24N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(u) 4)(d(v) 1)
3 3
4 4
3 3 2 2uv
2 2
d(u)d(v) d (u) d (v) 100d(u)d(v) 48
35d(u)d(v) d(u) d(v) d (u) d (v)24N =
10 d (u) d (v) 35 d (u) d (v)
74 d(u) d(v) 10d(u)d(v) d (u) d (v)
E(T)d(u,v) 2
3 3 4 4
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2 3 3
uv E(T)d(u,v) 2
d(u)d(v) d (u) d (v) 24N d (u) d (v)
10 d(u)d(v) d (u) d (v) 10 d (u) d (v)
uv E(T)d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
35 d(u)d(v) d(u) d(v) 35 d (u) d (v)
100 d(u)d(v) 74 d(u) d(v) 48N
2(P )
3 3
uv E(T)d(u,v) 2
iv. d(u)d(v) d (u) d (v) 24N 24N
60N
60N
50N 50N
4 3 230N(P ) 32N(P ) 2N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 6 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
4 3 2
uv E( )d(u,v) 2
i. d(u)d(v) N (P ) 2N (P ) N (P )
T T TT
4 3 2
xy E( )d(x,y) 2
d(x)d(y) N (P ) 2N (P ) N (P )
* * *T T T*T
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(x)d(y) d(u)d(v) 4
2
* T TT
L.H.S
3 2N (P ) 6N (P ) q T T
26VIT1984-2010
Creating Stars
4 3 2
uv E( )d(u,v) 2
ii. d(u)d(v) d(u) d(v) 2N 2N
6N (P ) 8N (P ) 2N (P )
T TT
T T T
4 3 2
xy E( )d(x,y) 2
d(x)d(y) d(x) d(y) 2N 2N
6N (P ) 8N (P ) 2N (P )
* *T T*T
* * *T T T
L.H.S
uv E( ) v V( )xy E( )
d(x,y) 2
d(v)d(x)d(y) d(x) d(y) d(u)d(v) d(u) d(v) 16
2
* T TT
1,3 3 22N 6N (K ) 6N (P ) 24N (P ) 2q
T T T T
26VIT1984-2010
Creating Stars
R.H.S
4 3 22N 2N 6N (P ) 8N (P ) 2N (P )
* * * * *T T T T T
1,3 3 22N 6N (K ) 6N (P ) 24N (P ) 2q
T T T T
N N
* TT
4 3 2N (P ) N (P ) N (P ) * T T
T
,
26VIT1984-2010
Creating Stars
2 2
uv E( )d(u,v) 2
iii. d(u)d(v) d (u) d (v) 6N 6N
12N
T TT
T
4 3 2
12N
14N (P ) 16N (P ) 2N (P )
T
T T T
2 2
xy E( )d(x,y) 2
d(x)d(y) d (x) d (y) 6N 6N
12N
* *T T*
T
*T
4 3 2
12N
14N (P ) 16N (P ) 2N (P )
*T
* * *T T T
26VIT1984-2010
Creating Stars
2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(x)d(y) d (x) d (y) d(u)d(v) d (u) d (v) 32
2
* T TT
R.H.S
L.H.S
1,4 1,3
3 2
6N 12N 24N (K ) 36N (K )
14N (P ) 48N (P ) 2q
T T T T
T T
4 3
6N 6N 12N
12N 14N (P ) 16N (P ) 2N
* * *T T T
* * *T T T
2(P )*
T
N N
* TT
1,4 1,3
3 2
6N 12N 24N (K ) 36N (K )
14N (P ) 48N (P ) 2q
T T T T
T T
26VIT1984-2010
Creating Stars
3 3
uv E( )d(u,v) 2
iv. d(u)d(v) d (u) d (v) 24N 24N
60N
T TT
T60N
50N 50N
T
T T
4 3 230N (P ) 32N (P ) 2N (P )
T T T
3 3
xy E( )d(x,y) 2
d(x)d(y) d (x) d (y) 24N 24N
60N
* *T T*
T
*T
60N
50N 50N
*T
* *T T
4 3 230N (P ) 32N (P ) 2N (P )
*T* *
T T
26VIT1984-2010
Creating Stars
3 3 3 3
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(x)d(y) d (x) d (y) d(u)d(v) d (u) d (v) 64
2
* T TT
R.H.S
L.H.S
1,5 1,4 1,3 3 2
24N 60N 50N
120N (K ) 240N (K ) 150N (K ) 30N (P ) 96N (P ) 2q
T T T
T T T T T
24N 24N 60N
60N 50N
* * *T T T
* *T T
4 3 2
50N
30N (P ) 32N (P ) 2N (P )
*T
* * *T T T
26VIT1984-2010
Creating Stars
N N
* TT
1,5 1,4 1,3 3 2
24N 60N 50N
120N (K ) 240N (K ) 150N (K ) 30N (P ) 96N (P ) 2q
T T T
T T T T T
102
26VIT1984-2010
Creating Stars
2 2
uv E(T)d(u,v) 2
4
i. d (u)d (v) 4N 6N
2N 9N(P ) 6N
3 2(P ) N(P )
Proposition 7: Let T be a tree with ‘q’ edges
2 2
uv E(T)d(u,v) 2
ii. d (u)d (v) d(u) d(v) 12N 48N
18N
50N
14N 6N
4 3 242N(P ) 20N(P ) 2N(P )
2 2 2 2
uv E(T)d(u,v) 2
iii. d (u)d (v) d (u) d (v) 48N 120N
200
N 72N
24N 60N
180N 180N
4 3 2
52N
90N(P ) 36N(P ) 2N(P )
3 3
uv E(T)d(u,v) 2
iv. d (u)d (v) 36N 72N 42N
84N
4
144N 6N
12N 49N(P ) 14N(P
3 2) N(P )
26VIT1984-2010
Creating Stars
Proof of Proposition 7
uv E(T)d(u,v) 2
d(u) 1 d(v) 1i. N =
2 2
uv E(T)d(u,v) 2
4N = (d(u) 1)(d(u) 2)(d(v) 1)(d(v) 2)
2 2
2 2
uv E(T)d(u,v) 2
d (u)d (v) 3d(u)d(v) d(u) d(v)
4N = 2 d (u) d (v) 6 d(u) d(v)
9d(u)d(v) 4
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d (u)d (v) 4N 9 d(u)d(v) 6 d(u) d(v)
2 d (u) d (v) 3 d(u)d(v)
2d(u) d(v) 4N(P )
2 2
uv E(T)d(u,v) 2
4
i. d (u)d (v) 4N 6N
2N 9N(P ) 6N
3 2(P ) N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1 d(u) 1 d(v) 1ii. N
2 3 3 2
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)(d(v) 1)(d(v) 2)(d(v) 3)12N
(d(u) 1)(d(u) 2)(d(u) 3)(d(v) 1)(d(v) 2)
2 2
2 2 3 3
2 2u
2 2
d (u)d (v) d(u) d(v) 40 d(u) d(v)
18 d (u) d (v) 2 d (u) d (v)12N
29d(u)d(v) d(u) d(v) 12d (u)d (v)
3d(u)d(v) d (u) d (v) 66d(u)d(v) 24
v E(T)d(u,v) 2
2 2 3 3
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2 2 2
uv E(T) uv E(d(u,v) 2
d (u)d (v) d(u) d(v) 12 2 d (u) d (v)
3 d(u)d(v) d (u) d (v) 12 d (u)d (v)
T)d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
2
uv E(T) uv E(T)d(u ,v) 2 d(u,v) 2
29 d(u)d(v) d(u) d(v) 66 d(u)d(v)
18 d (u) d (v) 40 d(u) d(v) 24N(P )
2 2
uv E(T)d(u,v) 2
ii. d (u)d (v) d(u) d(v) 12N 48N
18N
50N
14N 6N
4 3 242N(P ) 20N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1 d(u) 1 d(v) 1iii. N =
2 4 4 2
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)(d(v) 1)(d(v) 2)(d(v) 3)(d(v) 4)48N =
(d(u) 1)(d(u) 2)(d(u) 3)(d(u) 4)(d(v) 1)(d(v) 2)
2 2 2 2 4 4
3 3 3 3
2 2 2 2
2 2
d (u)d (v) d (u) d (v) 2 d (u) d (v)
3d(u)d(v) d (u) d (v) 20 d (u) d (v)
10d (u)d (v) d(u) d(v) 94 d (u) d (v)48N =
30d(u)d(v) d (u) d (v) 172 d(u
uv E(T)d(u,v) 2
2 2
) d(v)
155d(u)d(v) d(u) d(v) 300d(u)d(v)
70d (u)d (v) 96
26VIT1984-2010
Creating Stars
2 2 2 2 4 4
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
3 3 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d (u)d (v) d (u) d (v) 48N 2 d (u) d (v)
20 d (u) d (v) 94 d (u) d (v) 172 d(
uv E(T)d(u,v) 2
2 2 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
3 3
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
u) d(v)
10 d (u)d (v) d(u) d(v) 30 d(u)d(v) d (u) d (v)
155 d(u)d(v) d(u) d(v) 3 d(u)d(v) d (u) d (v)
2 2
2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
300 d(u)d(v) 70 d (u)d (v) 96N(P )
26VIT1984-2010
Creating Stars
2 2 2 2
uv E(T)d(u,v) 2
iii. d (u)d (v) d (u) d (v) 48N 120N
200
N 72N
24N 60N
180N 180N
4 3 2
52N
90N(P ) 36N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 1iv. N =
3 3
uv E(T)d(u,v) 2
36N = (d(u) 1)(d(u) 2)(d(u) 3)(d(v) 1)(d(v) 2)(d(v) 3)
3 3 2 2
2 2 3 3
2 2
2 2
d (u)d (v) 36d (u)d (v) 121d(u)d(v)
11d(u)d(v) d (u) d (v) 6 d (u) d (v)36N =
66d(u)d(v) d(u) d(v) 36 d (u) d (v)
6d (u)d (v) d(u) d(v) 66 d(u) d(v) 36
uv E(T)d(u,v) 2
26VIT1984-2010
Creating Stars
3 3 2 2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
2 2 3 3
uv E(T) uv E(d(u ,v) 2
d (u)d (v) 36N 36 d (u)d (v) 121 d(u)d(v)
11 d(u)d(v) d (u) d (v) 6 d (u) d (v)
T)d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
66 d(u)d(v) d(u) d(v) 36 d (u) d (v)
6 d (u)d (v) d(u) d(v) 66 d(u) d(v) 36N(P )
26VIT1984-2010
Creating Stars
3 3
uv E(T)d(u,v) 2
iv. d (u)d (v) 36N 72N 42N
84N
4
144N 6N
12N 49N(P ) 14N(P
3 2) N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 7 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
2 2
4
uv E( )d(u,v) 2
i. d (u)d (v) 4N 6N
2N 9N (P )
T TT
T T 3 26N (P ) N (P )T T
2 2
4
xy E( )d(x,y) 2
d (x)d (y) 4N 6N
2N 9N (P ) 6N (P
* *T T*T
* * *T T T
3 2) N (P )*
T
26VIT1984-2010
Creating Stars
2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x)d (y) d (u)d (v) 16
2
* T TT
R.H.S
L.H.S
1,3
3 2
4N 6N 6N (K )
9N (P ) 22N (P ) q
T T T
T T
4 3 2
4N 6N 2N
9N (P ) 6N (P ) N (P )
* * *T T T
* * *T T T
N N
* TT
1,3
3 2
4N 6N 6N (K )
9N (P ) 22N (P ) q
T T T
T T
26VIT1984-2010
Creating Stars
2 2
uv E( )d(u,v) 2
ii. d (u)d (v) d(u) d(v) 12N 48N
18N
T TT
T50N
14N 6N
T
T T
4 3 242N (P ) 20N (P ) 2N (P )
T T T
2 2
xy E( )d(x,y) 2
d (x)d (y) d(x) d(y) 12N 48N
18N
* *T T*
T
*T
50N
14N 6N
*T
* *T T
4 3 242N (P ) 20N (P ) 2N (P ) * * *
T T T
26VIT1984-2010
Creating Stars
2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x)d (y) d(x) d(y) d (u)d (v) d(u) d(v) 64
2
* T TT
R.H.S
L.H.S
1,4 1,3
12N 18N
48N 50N
24N (K ) 42N (K ) 42N
T T
T T
T T 3 2(P ) 84N (P ) 2q T T
12N 48N 18N
50N 14N
* * *T T T
* *T T
4 3 2
6N
42N (P ) 20N (P ) 2N (P )
*T
* * *T T T
26VIT1984-2010
Creating Stars
N N
* TT
1,4 1,3
12N 18N
48N 50N
24N (K ) 42N (K ) 42N
T T
T T
T T 3 2(P ) 84N (P ) 2q T T
2 2 2 2
uv E( )d(u,v) 2
iii. d (u)d (v) d (u) d (v) 48N
120N 200N 72N
TT
T T T
24N 60N 180N
180N
T T T
T
4 3 2
52N
90N (P ) 36N (P ) 2N (P )
T
T T T
26VIT1984-2010
Creating Stars
2 2 2 2
xy E( )d(x,y) 2
d (x)d (y) d (x) d (y) 48N
120N 200N 72N
*T*T
* * *T T T
24N 60N 180N
1
* * *T T T
4 3 2
80N 52N
90N (P ) 36N (P ) 2N (P )
* *T T
* * *T T T
2 2 2 2 2 2 2 2
uv E( )xy E( )d(x,y) 2
v V( )
d (x)d (y) d (x) d (y) d (u)d (v) d (u) d (v)
d(v) 128
2
* TT
T
L.H.S
26VIT1984-2010
Creating Stars
48N 72N 180N
180N 200N
T T T
T T
1,5 1,4 1,3 3 2
120N
120N (K ) 240N (K ) 156N (K ) 90N (P ) 164N (P ) 2q
T
T T T T T
48N 120N 200N
72N 24N
* * *T T T
* *T T
60N
180N 180N 52N
*T
* * *T T T
4 3 290N (P ) 36N (P ) 2N (P )
* * *
T T T
R.H.S
26VIT1984-2010
Creating Stars
N N
* TT
48N 72N 180N
180N 200N
T T T
T T
1,5 1,4 1,3 3 2
120N
120N (K ) 240N (K ) 156N (K ) 90N (P ) 164N (P ) 2q
T
T T T T T
26VIT1984-2010
Creating Stars
3 3
xy E( )d(x,y) 2
d (x)d (y) 36N 72N 42N
84N
* * *T T T*
T
*T
4
144N 6N
12N 49N (P ) 14N
* *T T
* *T T
3 2(P ) N (P )* *
T T
3 3
uv E(T)d(u,v) 2
iv. d (u)d (v) 36N 72N 42N
84N
T T T
T
4
144N 6N
12N 49N (P )
T T
T T 3 214N (P ) N (P )T T
26VIT1984-2010
Creating Stars
3 3 3 3
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (x)d (y) d (u)d (v) 64
2
* T TT
R.H.S
L.H.S
36N 72N 42N
84N 144N
T T T
T T
1,4 1,3 3 224N (K ) 36N (K ) 49N (P ) 78N (P ) q
T T T T
36N 72N 42N
84N 144N
* * *T T T
* *T T
4 3 2
6N
12N 49N (P ) 14N (P ) N (P )
*T
* * * *T T T T
26VIT1984-2010
Creating Stars
N N
* TT
36N 72N 42N
84N 144N
T T T
T T
1,4 1,3 3 224N (K ) 36N (K ) 49N (P ) 78N (P ) q
T T T T
125
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
(d(u) 1)(d(w) 2)32. N
(d(w) 2)(d(v) 1)
uv E(T)d(u,v) 2
d(u) 1 d(v) 133. N (d(w) 2) (d(w) 2)
2 2
(2,2)
1N N T
4
(1,2) (2,1)
1N N T T
2
(3,1) (1,3)
1N N T T
3
uv E(T)d(u,v) 2
d(w) 2 d(w) 234. N (d(u) 1) (d(v) 1)
2 2
126
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(w) 2 d(w) 2 d(v) 135. N
2 2 2 2
(3,2) (2,3)
1N N T T
3
Proposition 8: Let T be a tree with ‘q’ edges.
uw,wv E(T)
1,3 3 2
uv E(T)d(u,v) 2
i. d(w) d(u) d(v) 2N 6N(K ) 4N(P ) 4N(P )
2 2
uv E(T)d(u,v) 2
ii. d(w) d (u) d (v) 8N 10N
6N
1,3 3 2(K ) 12N(P ) 4N(P )
127
26VIT1984-2010
Creating Stars
2
uv E(T)d(u,v) 2
1,4 1,3
iii. d (w) d(u) d(v) 6N 10N
24N(K ) 30N(K )
3 28N(P ) 8N(P )
2 2 2
uv E(T)d(u,v) 2
iv. d (w) d (u) d (v) 40N 12N
18N
1,4 1,3 3 2
38N
24N(K ) 30N(K ) 24N(P ) 8N(P )
Proof of Proposition 8
uv E(T)d(u,v) 2
1i. N (d(w) 2)(d(v) 1) (d(w) 2)(d(u) 1)
2
uv E(T)d(u,v) 2
2N d(w) d(u) d(v) 2 d(u) d(v) 2d(w) 4
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2
uv E(T)d(u,v) 2
d(w) d(u) d(v) 2N 2 d(u) d(v)
2 d(w) 4N(P )
1,3 3 2
uv E(T)d(u,v) 2
i. d(w) d(u) d(v) 2N 6N(K ) 4N(P ) 4N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 11ii. N (d(w) 2) (d(w) 2)
2 24
uv E(T)d(u,v) 2
(d(w) 2)(d(u) 1)(d(u) 2)8N
(d(w) 2)(d(v) 1)(d(v) 2)
2 2
2 2uv E(T)d(u,v) 2
d(w) d (u) d (v) 3d(w) d(u) d(v)8N
4d(w) 2 d (u) d (v) 6 d(u) d(v) 8
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T) uv E(d(u,v) 2
d(w) d (u) d (v) 8N 3 d(w) d(u) d(v)
2 d (u) d (v) 6 d(u) d(v)
T)
d(u,v) 2
2
uv E(T)d(u,v) 2
4 d(w) 8N(P )
2 2
uv E(T)d(u,v) 2
ii. d(w) d (u) d (v) 8N 10N
6N
1,3 3 2(K ) 12N(P ) 4N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(w) 2 d(w) 21iii. N (d(u) 1) (d(v) 1)
2 23
uv E(T)d(u,v) 2
(d(w) 2)(d(w) 3)(d(u) 1)6N
(d(w) 2)(d(w) 3)(d(v) 1)
2
2uv E(T)d(u,v) 2
d (w) d(u) d(v) 5d(w) d(u) d(v)6N
6 d(u) d(v) 2d (w) 10d(w) 12
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
uv E(T) uvd(u,v) 2
d (w) d(u) d(v) 6N 6 d(u) d(v) 2 d (w)
5 d(w) d(u) d(v) 10 d(w)
2
E(T)d(u,v) 2
12N(P )
2
uv E(T)d(u,v) 2
1,4 1,3
iii. d (w) d(u) d(v) 6N 10N
24N(K ) 30N(K )
3 28N(P ) 8N(P )
uv E(T)d(u,v) 2
d(w) 2 d(u) 1 d(w) 2 d(v) 11iv. N
2 2 2 23
uv E(T)d(u,v) 2
(d(w) 2)(d(w) 3)(d(u) 1)(d(u) 2)12N
(d(w) 2)(d(w) 3)(d(v) 1)(d(v) 2)
2 2 2 2
2 2
2 2uv E(T)d(u,v) 2
2
d (w) d (u) d (v) 3d (w) d(u) d(v)
5d(w) d (u) d (v) 15d(w) d(u) d(v)12N
6 d (u) d (v) 18 d(u) d(v)
4d (w) 20d(w) 24
2 2 2 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,
d (w) d (u) d (v) 12N 6 d (u) d (v)
18 d(u) d(v) 15 d(w) d(u) d(v) 3 d (w) d(u) d(v)
v) 2
2 2 2
2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u ,v) 2
5 d(w) d (u) d (v) 4 d (w) 20 d(w) 24N(P )
2 2 2
uv E(T)d(u,v) 2
iv. d (w) d (u) d (v) 40N 12N
18N
1,4 1,3 3 2
38N
24N(K ) 30N(K ) 24N(P ) 8N(P )
26VIT1984-2010
Creating Stars
26VIT1984-2010
Creating Stars
Verification of Proposition 8 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
1,3 3 2
uv E( )d(u,v) 2
i. d(w) d(u) d(v) 2N 6N (K ) 4N (P ) 4N (P )
T T T T
T
1,3 3 2
xy E( )d(x,y) 2
d(z) d(x) d(y) 2N 6N (K ) 4N (P ) 4N (P )
* * * *
T T T T*T
R.H.S
L.H.S
xy E( ) v V( )xy E( )
d(x,y) 2
d(v)d(z) d(x) d(y) 2 d(u) d(v) 4 d(v)
2
* T TT
1,3 212N (K ) 12N (P ) 4q T T
26VIT1984-2010
Creating Stars
R.H.S
1,3 3 22N 6N (K ) 4N (P ) 4N (P )
* * * *T T T T
1,3 212N (K ) 12N (P ) 4q T T
2 2
uv E( )d(u,v) 2
ii. d(w) d (u) d (v) 8N 10N
TTT
1,3 3 26N (K ) 12N (P ) 4 N(P ) T T T
2 2
1,3
xy E( )d(x,y) 2
d(z) d (x) d (y) 8N 10N
6N (K
* *T T*T
*T
3 2) 12N (P ) 4N (P ) * *
T T
26VIT1984-2010
Creating Stars
2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(z) d (x) d (y) 2 d (u) d (v) 8 d(v)
2
* T TT
L.H.S
1,3 236N (K ) 28N (P ) 4q T T
1,3 3 28N 10N 6N (K ) 12N (P ) 4N (P )
* * * * *T T T T T
R.H.S
1,3 236N (K ) 28N (P ) 4q T T
26VIT1984-2010
Creating Stars
2
1,4 1,3
uv E( )d(u,v) 2
iii. d (w) d(u) d(v) 6N 10N
24N (K ) 30N (K ) 8N (
T TT
T T T 3 2P ) 8N (P )T
2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (z) d(x) d(y) 4 d(u) d(v) 4 d (v)
2
* T TT
2
1,4 1,3 3
xy E( )d(x,y) 2
d (z) d(x) d(y) 6N 10N
24N (K ) 30N (K ) 8N (P ) 8N
* *T T*T
* * *T T T T
2(P )*
L.H.S
1,4 1,3 248N (K ) 60N (K ) 24N (P ) 8q T T T
26VIT1984-2010
Creating Stars
R.H.S
1,4 1,3
3 2
6N 10N 24N (K ) 30N (K )
8N (P ) 8N (P )
* * * *T T T T
* *T T
1,4 1,3 248N (K ) 60N (K ) 24N (P ) 8q T T T
2 2 2
uv E( )d(u,v) 2
iv. d (w) d (u) d (v) 40N 12N
18N
T TT
T
1,4 1,3 3 2
38N
24N (K ) 30N (K ) 24N (P ) 8N (P )
T
T T T T
26VIT1984-2010
Creating Stars
2 2 2
xy E( )d(x,y) 2
d (z) d (x) d (y) 40N 12N
18N
* *T T*
T
*T
1,4 1,3 3 2
38N
24N (K ) 30N (K ) 24N (P ) 8N (P )
*T
* * * *T T T T
2 2 2 2 2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d (z) d (x) d (y) 4 d (u) d (v) 8 d (v)
2
* T TT
L.H.S
1,4 1,3 296N (K ) 144N (K ) 56N (P ) 8q T T T
26VIT1984-2010
Creating Stars
1,4 1,3
40N 12N
18N 38N
24N (K ) 30N (K ) 24N (
* *T T
* *T T
* * *T T T
3 2P ) 8N (P )*
T
R.H.S
1,4 1,3 296N (K ) 144N (K ) 56N (P ) 8q T T T
142
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(w) 237. N (d(u) 1) (d(v) 1)
2
uv E(T)d(u,v) 2
d(v) 1d(u) 1 (d(w) 2)
238. N
d(u) 1(d(w) 2) d(v) 1
2
uv E(T)d(u,v) 2
d(w) 2 d(v) 1d(u) 1
2 239. N
d(v) 1 d(w) 2d(v) 1
2 2
uv E(T)d(u,v) 2
36. N (d(u) 1)(d(w) 2)(d(v) 1)
143
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(v) 140. N (d(w) 2)
2 2
uv E(T)d(u,v) 2
d(u) 1 d(w) 2 d(v) 141. N
2 2 2
Proposition 9: Let T be a tree with ‘q’ edges.
uw,wv E(T)
uv E(T)d(u,v) 2
1,3 4 3 2
i. d(u)d(w)d(v) N 2N
3N(K ) 2N(P ) 4N(P ) 2N(P )
144
26VIT1984-2010
Creating Stars
2
uv E(T)d(u,v) 2
ii. d(u)d (w)d(v) 2N 5N 6N
10N
1,4 1,3 4
3 2
12N(K ) 15N(K ) 4N(P )
8N(P ) 4N(P )
uv E(T)d(u,v) 2
iii. d(u)d(w)d(v) d(u) d(v) 2N 4N
6N 12N
1,3 4 3 2
8N
6N(K ) 12N(P ) 16N(P ) 4N(P )
145
26VIT1984-2010
Creating Stars
2
uv E(T)d(u,v) 2
iv. d(u)d (w)d(v) d(u) d(v)
4N 12N 30N
8N
10N 24N
48N 40N 12N
1,4 1,3 4 3 224N(K ) 30N(K ) 24N(P ) 32N(P ) 8N(P )
26VIT1984-2010
Creating Stars
Proof of Proposition 9
uv E(T)d(u,v) 2
i. N (d(u) 1)(d(w) 2)(d(v) 1)
uv E(T)d(u,v) 2
d(w) d(w) d(u) d(v) d(u)d(w)d(v)N
2 2 d(u) d(v) 2d(u)d(v)
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,
d(u)d(w)d(v) N d(w) d(w) d(u) d(v)
2 d(u) d(v) 2 d(u)d(v)
2
v) 2
2N(P )
uv E(T)d(u,v) 2
1,3 4 3 2
i. d(u)d(w)d(v) N 2N
3N(K ) 2N(P ) 4N(P ) 2N(P )
uv E(T)d(u,v) 2
d(w) 2ii. N (d(u) 1) (d(v) 1)
2
uv E(T)d(u,v) 2
2N (d(u) 1)(d(w) 2)(d(w) 3)(d(v) 1)
2
uv E(T)d(u,v) 2
1 d(u) d(v) d(u)d(v)2N
d (w) 5d(w) 6
2 2 2
uv E(T)d(u,v) 2
d (w) d (w) d(u) d(v) d(u)d (w)d(v)
2N 5d(w) 5d(w) d(u) d(v) 5d(u)d(w)d(v)
6 6 d(u) d(v) 6d(u)d(v)
26VIT1984-2010
Creating Stars
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
d(u)d (w)d(v) 2N d (w) d(u) d(v)
d (w) 5 d(w) 5 d(w) d(u) d(v)
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2
uv E(T)d(u,v) 2
5 d(u)d(w)d(v) 6 d(u) d(v)
6 d(u)d(v) 6N(P )
2
uv E(T)d(u,v) 2
ii. d(u)d (w)d(v) 2N 5N 6N
10N
1,4 1,3 4
3 2
12N(K ) 15N(K ) 4N(P )
8N(P ) 4N(P )
uv E(T)d(u,v) 2
d(v) 1(d(u) 1)(d(w) 2)
2iii. N
d(u) 1(d(w) 2)(d(v) 1)
2
uv E(T)d(u,v) 2
(d(u) 1)(d(w) 2)(d(v) 1)(d(v) 2)2N
(d(u) 1)(d(u) 2)(d(w) 2)(d(v) 1)
2 2
2 2uv E(T)d
d(u)d(w)d(v) d(u) d(v) d(w) d (u) d (v)
2d(u)d(v) d(u) d(v) 10 d(u) d(v)2N
2 d (u) d (v) 6d(u)d(w)d(v) 4d(w)
5d(w) d(u) d(v) 12d(u)d(v) 8
(u,v) 2
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u ,v) 2
uv E(T) uv E(T)d(u,v) 2 d
d(u)d(w)d(v) d(u) d(v) 2N d(w) d (u) d (v)
2 d(u)d(v) d(u) d(v) 10 d(u) d(v)
(u,v) 2
2 2
2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 d (u) d (v) 6 d(u)d(w)d(v) 8N(P )
4 d(w) 5 d(w) d(u) d(v) 12
uv E(T)d(u,v) 2
d(u)d(v)
uv E(T)d(u,v) 2
iii. d(u)d(w)d(v) d(u) d(v) 2N 4N
6N 12N
1,3 4 3 2
8N
6N(K ) 12N(P ) 16N(P ) 4N(P )
uv E(T)d(u,v) 2
(d(u) 1)(d(w) 2)(d(w) 3)(d(v) 1)(d(v) 2)4N
(d(u) 1)(d(u) 2)(d(w) 2)(d(w) 3)(d(v) 1)
uv E(T)d(u,v) 2
d(w) 2 d(v) 1(d(u) 1)
2 2iv. N
d(u) 1 d(w) 2(d(v) 1)
2 2
2 2
2 2 2 2 2
2 2
d(u)d (w)d(v) d(u) d(v) 6d(u)d (w)d(v)
5d (w) d(u) d(v) d (w) d (u) d (v) 4d (w)
5d(w)d(u)d(v) d(u) d(v) 30d(u)d(w)d(v)4N
25d(w) d(u) d(v) 5d(w) d (u) d (v) 20d(
uv E(T)d(u,v) 2
2 2
w)
6d(u)d(v) d(u) d(v) 36d(u)d(v) 30 d(u) d(v)
6 d (u) d (v) 24
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d(u)d (w)d(v) d(u) d(v) 4N 6 d(u)d (w)d(v)
5 d(u)d(w)d(v) d(u) d(v) 30 d(u)d(w)d(v) 36 d(u
uv E(T)d(u,v) 2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
2 2 2
uv E(T) uv E(Td(u,v) 2
)d(v)
25 d(w) d(u) d(v) 6 d(u)d(v) d(u) d(v) 30 d(u) d(v)
6 d (u) d (v) 5 d (w) d(u) d(v)
2 2 2
) uv E(T)d(u,v) 2 d(u,v) 2
2 2 2
2
uv E(T) uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
d (w) d (u) d (v)
5 d(w) d (u) d (v) 4 d (w) 20 d(w) 24N(P )
26VIT1984-2010
Creating Stars
2
uv E(T)d(u,v) 2
iv. d(u)d (w)d(v) d(u) d(v)
4N 12N 30N
8N
10N 24N
48N 40N 12N
1,4 1,3 4 3 224N(K ) 30N(K ) 24N(P ) 32N(P ) 8N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 9 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
1,3 4 3 2
uv E( )d(u,v) 2
i. d(u)d(w)d(v) N 2N
3N (K ) 2N (P ) 4N (P ) 2N (P )
T TT
T T T T
1,3 4 3 2
xy E( )
d(x,y) 2
d(x)d(z)d(y) N 2N
3N (K ) 2N (P ) 4N (P ) 2N (P )
* *T T*
T
* * * *T T T T
26VIT1984-2010
Creating Stars
L.H.S
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(x)d(z)d(y) 2d(u)d(v) 4 d(v)
2
* T TT
1,3 3 212N (K ) 2N (P ) 12N (P ) 2q T T T
R.H.S
1,3 4 3 2
N 2N
3N (K ) 2N (P ) 4N (P ) 2N (P )
* *T T
* * * *T T T T
1,3 3 212N (K ) 2N (P ) 12N (P ) 2q T T T
26VIT1984-2010
Creating Stars
2
uv E( )d(u,v) 2
ii. d(u)d (w)d(v) 2N 5N 6N
10N
T T TT
1,4 1,3 4
3 2
12N (K ) 15N (K ) 4N (P )
8N (P ) 4N (P )
T T T T
T T
2
xy E( )d(x,y) 2
d(x)d (z)d(y) 2N 5N 6N
10N
* * *T T T*
T
*T
1,4 1,3 4
3 2
12N (K ) 15N (K ) 4N (P )
8N (P ) 4N (P )
* * *T T T
* *T T
26VIT1984-2010
Creating Stars
2 2
uv E( ) v V( )xy E( )d(x,y) 2
d(v)d(x)d (z)d(y) 4d(u)d(v) 4 d (v)
2
* T TT
L.H.S
R.H.S
1,4 1,3 3 248N (K ) 60N (K ) 4N (P ) 24N (P ) 4q T T T T
1,4 1,3
2N 5N 6N
10N 12N (K ) 15N (K )
4N
* * *T T T
* * *T T T
T4 3 2(P ) 8N (P ) 4N (P )
* * *T T
1,4N 6N (K )
* TT
1,4 1,3 3 248N (K ) 60N (K ) 4N (P ) 24N (P ) 4q T T T T
26VIT1984-2010
Creating Stars
uv E( )d(u,v) 2
iii. d(u)d(w)d(v) d(u) d(v) 2N 4N
6N 12N
T TT
T T
1,3 4 3 2
8N
6N (K ) 12N (P ) 16N (P ) 4N (P )
T
T T T T
xy E( )d(x,y) 2
d(x)d(z)d(y) d(x) d(y) 2N 4N
6N 12N
* *T T*
T
* *T T
1,3 4 3 2
8N
6N (K ) 12N (P ) 16N (P ) 4N (P )
*T
* * * *T T T T
26VIT1984-2010
Creating Stars
L.H.S
R.H.S
uv E( )xy E( )
d(x,y) 2
v V( )
d(x)d(z)d(y) d(x) d(y) 2d(u)d(v) d(u) d(v)
d(v) 16 d(v)
2
* TT
T
1,3 3 24N 60N (K ) 12N (P ) 48N (P ) 4q
T T T T
2N 4N 6N
12N 8N
* * *T T T
* *T T
1,3 4 3 26N (K ) 12N (P ) 16N (P ) 4N (P )
* * * *
T T T T
26VIT1984-2010
Creating Stars
N 0
*T
1,3 3 24N 60N (K ) 12N (P ) 48N (P ) 4q
T T T T
2
uv E( )d(u,v) 2
iv. d(u)d (w)d(v) d(u) d(v)
4N 12N 30N
8N
T
T T T
T 10N 24N
48N 40N 12
T T
T T
1,4 1,3 4 3 2
N
24N (K ) 30N (K ) 24N (P ) 32N (P ) 8N (P )
T
T T T T T
26VIT1984-2010
Creating Stars
2
xy E( )d(u,v) 2
d(x)d (z)d(y) d(x) d(y)
4N 12N 30N
8N
*
T
* * *T T T
*T10N 24N
48N 40N 12N
* *T T
* * *T T T
1,4 1,3 4 3 224N (K ) 30N (K ) 24N (P ) 32N (P ) 8N (P )
* * * * *T T T T T
2
uv E( )xy E( )d(u,v) 2
2
v V( )
d(x)d (z)d(y) d(x) d(y) 4d(u)d(v) d(u) d(v)
d(v) 16 d (v)
2
* TT
T
L.H.S
26VIT1984-2010
Creating Stars
1,4 1,3
3 2
8N 192N (K ) 264N (K )
24N (P ) 96N (P ) 8q
T T T
T T
4N 12N 30N
8N 10N
* * *T T T
* *T T
1,4
24N
48N 40N 12N
24N (K ) 30N
*
*T
* * *T T T
T T 1,3 4 3 2(K ) 24N (P ) 32N (P ) 8N (P ) * * * *T T T
R.H.S
1,4 1,3
3 2
8N 192N (K ) 264N (K )
24N (P ) 96N (P ) 8q
T T T
T T
*TN 0
163
26VIT1984-2010
Creating Stars
Proposition 10: Let T be a tree with ‘q’ edges.
uw,wv E(T)
2 2
uv E(T)d(u,v) 2
i. d (u)d(w)d (v) 4N 6N
9N 8N
8N 10N 12N
1,3 4 3 23N(K ) 18N(P ) 12N(P ) 2N(P )
164
26VIT1984-2010
Creating Stars
2 2 2
uv E(T)d(u,v) 2
ii. d (u)d (w)d (v)
8N 18N 20N
30N
45N 40N
16N 12N
38N
24N 18N 12N
1,4 1,3 4 3 212N(K ) 15N(K ) 36N(P ) 24N(P ) 4N(P )
26VIT1984-2010
Creating Stars
Proof of Proposition 10
uv E(T)d(u,v) 2
d(u) 1 d(v) 1i. N (d(w) 2)
2 2
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)
4N (d(w) 2)
(d(v) 1)(d(v) 2)
2
uv E(T)2
d(u,v) 2
d (u) 3d(u) 2
4N (d(w) 2)
d (v) 3d(v) 2
26VIT1984-2010
Creating Stars
2 2
2 2
2 2
2 2
d (u)d(w)d (v) 3d(u)d(w)d(v) d(u) d(v)
2d(w) d (u) d (v) 6d(w) d(u) d(v)
9d(u)d(w)d(v) 4d(w)
4N
2d (u)d (v) 6d(u)d(v) d(u) d(v)
4 d (u) d (v) 12 d(u) d(v)
18d(u)d
uv E(T)d(u,v) 2
(v) 8
26VIT1984-2010
Creating Stars
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
d (u)d(w)d (v) 4N 9 d(u)d(w)d(v)
3 d(u)d(w)d(v) d(u) d(v) 4 d(w)
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T)d(u,
2 d(w) d (u) d (v) 6 d(w) d(u) d(v)
2 d (u)d (v) 6 d(u)d(v) d(u) d(v)
4 d (u) d (v)
uv E(T)
v) 2 d(u,v) 2
2
uv E(T)d(u,v) 2
12 d(u) d(v)
18 d(u)d(v) 8N(P )
26VIT1984-2010
Creating Stars
2 2
uv E(T)d(u,v) 2
i. d (u)d(w)d (v) 4N 6N
9N 8N
8N 10N 12N
1,3 4 3 23N(K ) 18N(P ) 12N(P ) 2N(P )
26VIT1984-2010
Creating Stars
uv E(T)d(u,v) 2
d(u) 1 d(w) 2 d(v) 1ii. N
2 2 2
uv E(T)d(u,v) 2
(d(u) 1)(d(u) 2)
8N (d(w) 2)(d(w) 3)
(d(v) 1)(d(v) 2)
2
2 2
2 2uv E(T)d(u,v) 2
d (w) 5d(w) 6
d (u)d (v) 3d(u)d(v) d(u) d(v)8N
2 d (u) d (v) 6 d(u) d(v)
9d(u)d(v) 4
26VIT1984-2010
Creating Stars
2 2 2 2
2 2 2 2
2 2
2 2
2 2
d (u)d (w)d (v) 3d(u)d (w)d(v) d(u) d(v)
2d (w) d (u) d (v) 6d (w) d(u) d(v)
9d(u)d (w)d(v) 4d (w)
5d (u)d(w)d (v) 15d(u)d(w)d(v) d(u) d(v)
8N 10d(w) d (u) d (v)
uv E(T)d(u,v) 2
2 2
2 2
30d(w) d(u) d(v)
45d(u)d(w)d(v) 20d(w)
6d (u)d (v) 18d(u)d(v) d(u) d(v)
12 d (u) d (v) 36 d(u) d(v)
54d(u)d(v) 24
26VIT1984-2010
Creating Stars
2 2 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2 2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v)
8N d (u)d (w)d (v) 9 d(u)d (w)d(v)
2 d (w) d (u) d (v) 6 d (w) d(u) d(v)
2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
3 d(u)d (w)d(v) d(u) d(v) 4 d (w)
5 d (u)d(w)d (v) 15 d(u)d(w)d(v) d(u) d(v)
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
2 2
uv E(T)
10 d(w) d (u) d (v) 30 d(w) d(u) d(v)
45 d(u)d(w)d(v) 20 d(w)
6 d (u)d (v)
2 2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2 d(u,v) 2
2
uv E(T) uv E(T)d(u,v) 2 d(u,v) 2
18 d(u)d(v) d(u) d(v) 12 d (u) d (v)
36 d(u) d(v) 54 d(u)d(v) 24N(P )
26VIT1984-2010
Creating Stars
Verification of Proposition 10 Let T be a tree on ‘q’ edges
Let T* be the tree obtained from T by complete subdivision.
2 2
uv E( )d(u,v) 2
i. d (u)d(w)d (v) 4N 6N
9N
T T
T
T 8N
8N 10N 12N
T
T T T
1,3 4 3 23N (K ) 18N (P ) 12N (P ) 2N (P ) T T T T
26VIT1984-2010
Creating Stars
2 2
xy E( )d(x,y) 2
d (x)d(z)d (y) 4N 6N
9N
*
* *T TT
*T8N
8N 10N 12N
*T
* * *T T T
1,3 4 3 23N (K ) 18N (P ) 12N (P ) 2N (P ) * * * *T T T T
L.H.S
2 2 2 2
xy E( ) uv E( ) v V( )d(x,y) 2
d(v)d (x)d(z)d (y) 2 d (u)d (v) 16 d(v)
2
*T T T
1,3 3 28N 12N 60N (K ) 18N (P ) 44N (P ) 2q
T T T T T
26VIT1984-2010
Creating Stars
1,3 3 28N 12N 60N (K ) 18N (P ) 44N (P ) 2q
T T T T T
4N 6N
9N 8N
8N
* *T T
* *T T
*T
1,3 4 3 2
10N
12N
3N (K ) 18N (P ) 12N (P ) 2N (P )
*T
*T
* * * *T T T T
R.H.S
*TN 0
26VIT1984-2010
Creating Stars
2 2 2
uv E( )d(u,v) 2
ii. d (u)d (w)d (v)
8N 18N 20N
T
T T T
30N 45N 40N
16N 12N
T T T
T T 38N
24N 18N 12N
T
T T T
1,4 1,3 4 3 212N (K ) 15N (K ) 36N (P ) 24N (P ) 4N (P )
T T T T T
26VIT1984-2010
Creating Stars
2 2 2
xy E( )d(x,y) 2
ii. d (x)d (z)d (y)
8N 18N 20N
*
*
T
* * TT T
30N 45N 40N
16N 12N
* * *
* *
T T T
T T 38N
24N 18N 12N
*
* * *
T
T T T
1,4 1,3 4 3 212N (K ) 15N (K ) 36N (P ) 24N (P ) 4N (P )
* * * * *T T T T T
26VIT1984-2010
Creating Stars
2 2 2 2 2 2
xy E( ) uv E( ) v V( )d(x,y) 2
d(v)d (x)d (z)d (y) 4 d (u)d (v) 16 d (v)
2
*T T T
L.H.S
1,4 1,3
3 2
16N 24N 192N (K ) 264N (K )
36N (P ) 88N (P ) 4q
T T T T
T T
26VIT1984-2010
Creating Stars
8N 18N 20N
30N
*
*
* * TT T
T45N 40N
16N 12N 38N
* *
* *
T T
T T T
24N 18N 12N
*
* * *T T T
1,4 1,3 4 3 212N (K ) 15N (K ) 36N (P ) 24N (P ) 4N (P ) * * * * *T T T T T
R.H.S
1,4 1,3
3 2
16N 24N 192N (K ) 264N (K )
36N (P ) 88N (P ) 4q
T T T T
T T
*TN 0
26VIT1984-2010
Creating Stars
Theorem 1:
Let T be a tree on ‘q’ edges.
jm m
Let V( ) , , ,. . .,
Define H (m) as follows:
V H (m) V( ) , , ,. . .,
E H (m) , , , , , \{ } E( )
for 1 m q 1
i
T
T 1 2 3
1 2 3 q+1
i iT
q
j j
+1u
v v v v
v
u
v
T
T
v v , v
u
u u T
u
u
26VIT1984-2010
Creating Stars
3
1,3
3 2
xy E(H (m))
x N ( )
d(x)d(y) d(x) d(y)
=2 d( ) 1 4N 12N (K )
12N (P ) 16N (P ) 4q
4d( ) d(x) 2 d(x) d(x)
m
m
T
v
T
T
T T
T m
v
v x N ( )
1
T vm
1v 3
v2
v
4vExample
T
1v
4v
3v
2u
2v
3u
1u
4u
1v
4v
3v
2u
2v
3u
1u
4u
T
T
H m)T
(
26VIT1984-2010
Creating Stars
mv
mu
i1u
i2u
i3u
iru
i1v
i2v
i3v
irv
H mT
r d ( ) d ( ) m mT Tv u
26VIT1984-2010
Creating Stars
Proof of Theorem 1:
In the new tree H (m)
V H (m) 2 V( )
E H (m) 2 E( ) 1
, E( ) , , E H (m)
i j i
T
j i j
T
T
T
T
T
vu uvv v T
Let E H (m)
, ,
1 k d( ) d( )
m m m m
m m
T
iik k
U
U , v u,v u v
v u
u
iv jvT
ivjv
iu j
u
H m)T
(
26VIT1984-2010
Creating Stars
T T
T T
T
H (m) H (m) T
2d ( ) 1
d ( ) d ( ) d ( ) 1
E H (m) E H (m)
m
m m m
v
v u
U
v
U
\ U
xy E(H (m)) xyT
xy (H (m))T
d(x)d(y) d(x) d(y) d(x)d(y) d(x) d(y)
d(x)d(y) d(x) d(y)
U
\U
26VIT1984-2010
Creating Stars
3
xy E(H (m))T
d( )
k 1
xy (H (T
d(x)d(y) d(x) d(y) 2 d( ) 1
2 d( ) 1 d( ) d( ) d( ) 1
d(x)d(y) d(x) d(y)
i ik k
m
vm
m mu
v
v uv
m))
\U
2
Expanding d( ) 1 d( ) d( ) d( ) 1
d( )d( ) d( ) d( )
2d( )d( ) d ( ) d( )
i ik k
i ik k
i i ik k k
m m
m m
m
v v
v v
u uv
u u
u u
u
26VIT1984-2010
Creating Stars
3
xy E(H (m))T
d( )
k 1
d( ) d
k 1 k 1
d(x)d(y) d(x) d(y) 2 d( ) 1
2 d( )d( ) d( ) d( )
4 d( )d( ) 2 d(
i
m
v
ik k
i
m
m m
vm
m ik k
v
v u u
u u
v
v
(v )m
xy E(H (m))T
) d( ) 1
d(x)d(y) d(x) d(y)
ik
\U
u
26VIT1984-2010
Creating Stars
3
uv E(T)
x N ( ) x N ( )T T
2 d( ) 1 2 d(u)d(v) d(u) d(v)
4d( ) d(x) 2 d(x) d(x) 1
m
mv vm m
v
v
3
1,3
3 2
xy E(H (m))T
T T
T T
x N ( )T
d(x)d(y) d(x) d(y)
=2 d( ) 1 4N 12N (K )
12N (P ) 16N (P ) 4q
4d( ) d(x) 2 d(x)
m
mvm
v
v x N ( )T
d(x) 1
vm