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  • 26VIT1984-2010

    Creating Stars

  • 26VIT1984-2010

    Creating Stars

    A Place to Learn ; A Chance To Grow

  • Arithmetical properties of Tree Generation

    Codes and Algorithm to generate all Tree

    Codes for a given number of Edges

    N.Chandramowliswaran

    Applied Algebra Division

    School of Advanced Sciences

    VIT University

    26VIT1984-2010

    Creating Stars

    ncmowli@hotmail.com

    N.Chandramowliswaran

    Applied Algebra Division

    School of Advanced Sciences

    VIT University

    26VIT1984-2010

    Creating Stars

    ncmowli@hotmail.com

    Claude Berge Bill Tutte

    Generation of Graceful Trees through

    Graceful Codes

  • Abstract

    Graceful Code is a way to represent graceful graph in terms

    of sequence of non-negative integers. Given a graceful

    graph G on q edges, we can generate its graceful code in

    the form of (a1, a2, a3, ., aq-1, aq=0) to represent the

    graph. Similarly, we can easily draw the graph from the

    given graceful code.

    Graceful codes are classified into two categories, namely,

    -valuable code and gracious code based on their

    properties. Graceful code provides an useful and efficient

    techniques to study and analyze graphs using computer.

    Here we discuss generation of infinitely many graceful

    codes, -valuable codes and gracious codes for a given

    graceful code, -valuable code and a gracious code.

  • Introduction

    A simple graph G(V,E) on p vertices and q

    edges is said to be graceful if there exist

    an injection f: V{0, 1, 2,.,q} such that the

    induced function g: E{1, 2, 3, , q} which is

    defined by g(u, v)=|f(u)-f(v)| for every edge

    (u, v), is a bijective function; then f is called

    graceful labelling of G.

  • Graceful Code Let G be any graceful graph on q edges then

    (a1, a2, a3, , aq -1, aq) is called a graceful code of G,

    if 0 ai q - i; 1 i q.

    Here ai is the lower end vertex of the edge label i.

    It is important to note that aq is always zero

    For every graceful graph G we can write its code. Conversely, for every given graceful code we can draw the corresponding graceful graph as follows.

    Join edges:(a1,1+a1),(a2,2+a2),...,(aq - 1,q-1+aq-1), (aq, q+aq)

  • Example 1

    Figure 1 shows a graceful graph on q = 7 edges

    with edge labeled from 1 to 7.

    7

    0

    6 3

    1 2

    4 5

    Code = (4, 2, 3, 0, 1, 0, 0)

    Figure 1

  • - valuable Code

    A graceful code (a1, a2, a3,..., aq-1, aq) of a

    graceful graph G on q edges is called

    - valuable code if

    Here a1 is called the separator or critical value

    of the - valuable code.

    a1 ai

    Max{ai| 1 i q} < Min{i+ai| 1 i q}.

  • Proposition

    (a1, a2, a3, ,aq-1,aq) represents -valuable code if

    and only if

    0 (a1 aq - i + 1) / q i) 1 for all i , 1 i q - 1

    Equivalently (a1, a2, a3, ,aq-1,aq) represents an

    - valuable code if and only if

    (a1 - aq, a1 - aq -1, , a1 - a3, a1 - a2, 0) represents a

    code of a Graceful Graph.

  • Properties of Graceful Codes

    1.1 If (a1, a2, a3,, aq - 1, aq) represents a code of

    a graceful graph G on q edges,

    then, (a2, a3,, aq - 1, aq) represents a code

    of some graceful graph H on q - 1 edges.

    1.2 If (a1, a2, a3,, aq-1, aq) is an valuable

    code on q edges and (q -1 - a1 > a1)

    then (q 1 - a1, q 2 - a2, , 1- aq - 1, 0,

    a1, a2,a3,, aq) is an valuable code

    on 2q edges.

  • 1.3. If (a1, a2, a3,, aq - 1, aq) is an valuable

    code on q edges and (a1> q -1 - a1) then,

    (a1, a2, a3,, aq, q - 1- a1, q 2 - a2, ,1 aq - 1, 0)

    is an valuable code on 2q edges.

    1.4. If (a1, a2, a3,, aq1 - 1, aq1) and

    (b1, b2, b3, , bq2 - 1, bq2) represents

    valuable codes on q1 and q2 edges

    respectively and a1 b1 then,

    (a1, a2, a3,, aq1 - 1, aq1, b1, b2, b3,, bq2 - 1, bq2)

    represents an valuable code

    on q1 + q2 edges.

  • 1.5. Let (a1, a2, a3,, aq -1, aq) represents

    a graceful code of a graph G on

    q edges then,

    (aq+ q, aq - 1+ q - 1,, 2 + a2, 1 + a1, a1, a2, a3,, aq - 1, aq)

    represents a valuable code on 2q edges.

  • Properties of Graceful Codes

    If (a1, a2, a3,, aq - 1, aq) represents a graceful

    code of a graceful graph G on q edges then,

    (aq+ q, aq 1 + q - 1,,2 + a2, 1 + a1, x, a1, a2, a3, aq - 1, aq),

    [0 x q] represents an valuable code

    on 2q + 1 edges.

  • If (a1, a2, , aq 1, aq) represent a code of a

    graceful graph G on q edges,

    Then,

    (q aq, q aq 1, , q a2, q a1, a1, a2, , aq 1, aq)

    represent a -valuable code on 2 q edges.

  • Properties of Graceful Codes

    Let X1, X2, X3, , Xr represent r valuable

    codes on edges qi

    (1 i r) having separators si respectively.

    Then, r-1 r-2 r-3

    ( sj + Xr , sj + Xr-1 , sj + Xr-2 , s1+ s2+ X3, s1+ X2, X1)

    j=1 j=1 j=1

    r

    always represent a valuable code on qj edges.

    j = 1

  • Tree Generation Theorems

    Let G be any simple graph on n vertices and q

    edges.

    Define a bipartite graph HG as follows:

    (vi, vj) E(G) (vi, vj) E(HG) and

    (vi, vj) E(HG).

    Join any vk V(G) V(HG),

    [1 k n] to vk V(HG).

    Here |V(HG)| = 2|V(G)| and |E(HG)| = 2 | E(G)| +1.

  • Tree Generation Theorems

    Moreover if G has a code (a1, a2, a3,, aq - 1, aq)

    then HG has an valuable code

    (aq+ q, aq-1+q - 1,, 2+a2, 1 + a1, x, a1, a2, a3, , aq - 1, aq) [0 x q].

    If G happens to be a bipartite graph, then HG

    contains two copies of G

    together with an edge connecting vk to vk

  • Examples

    Code = (0, 1, 0, 0)

    G

    HG

    Code = (4, 3, 3, 1, 3, 0, 1, 0, 0)

  • Examples

    G

  • Construction of HG

  • ai i + ai

    i

    E (G)

    ai i + ai

    q+1+ai q+1+i+ai

    E(HG)

    q+1+i q+1-i

    (aq+q, ,ai+i, ,1+a1, x, a1, , ai, , aq)

    q+1+ai q+1+i+ai

    q+1-i q+1+i

    i i

  • Tree Generation Theorems

    Theorem

    If (a1, a2, a3,, aq -1, aq) represents a valuable

    code of some tree T . Then,

    (aq+q, aq - 1+q-1, ,2+a2, 1+a1, a1, a2, a3,, aq - 1, aq)

    represents a valuable code of a tree S

    on 2q edges such that

    E(S) = E(T) U E(T).

  • Tree Generation Theorems

    Theorem

    If (a1, a2, a3,, aq-1, aq) is an valuable code of a

    graceful graph G on q edges, then,

    (a1, a2, a3,, aq-1, aq) represents a tree if and only if

    (a2, a3,, aq -1, aq) represents a tree

    on q - 1 edges.

  • Tree Generation Theorems

    If (a1, a2, , aq-2, aq-1, aq ) represents a code of a

    graceful tree on q edges, then

    1. (qk - 1, ka1, (q 1) k 1, ka2, , 2

    k - 1, kaq - 1, 1k - 1, kaq)

    represent a tree code on kq edges (k 2).

    2. (qk - 1, ka1+r, (q 1) k 1, ka2+r, , 2

    k - 1, kaq-1+r, 1k - 1,

    kaq+r, 0r) ;1 r k, k 2 represent a tree code

    on kq+r edges.

  • Corollary 1

    If (a1, a2, , aq - 2, aq - 1, aq ) represents a code of a

    graceful tree on q edges, then

    (q, 2a1, q - 1, 2a2, q - 2, 2a3, , 2, 2aq - 1, 1, 2aq)

    represent a code of a graceful tree on 2q edges and (q, 2a1+1, q - 1, 2a2+1, q - 2, 2a3+1, , 2, 2aq - 1+1, 1,

    2aq+1,0) represent a tree code on 2q+1 edges.

  • Corollary 2

    If (a1, a2, , aq-2, aq-1, aq ) represents a code of a

    graceful tree on q edges, then

    (q+1, 2a1, q, 2a2, q - 1, 2a3, , 3, 2aq-1, 2, 2aq, 1, 0)

    represent a code of a graceful tree on

    2q+2 edges and (q+1, 2a1+1, q, 2a2+1, q - 1,

    2a3+1, , 3, 2aq - 1+1, 2, 2aq+1, 1, 0, 0)

    represent a tree code on 2q + 3 edges.

  • Tree Generation Theorems

    Using - valuable tree codes

    Theorem 1

    If (a1, a2, , aq - 1, aq) represent a -valuable tree

    code on q edges, then,

    (aq+q, aq - 1+ q 1, , 2 + a2, 1+ a1, 1 + a1, a1, a1,a2,

    , aq - 1, aq)

    represent a -valuable tree code on 2q+2 edges.

  • Theorem 2

    Let (a1, a2, , aq1 - 2, aq1 - 1, aq1) represents a

    - valuable tree code on q1 edges and

    (b1, b2, , bq2 - 2, bq2 - 1, bq2) represent a tree code on

    q2 edges. Then,

    1. (a1 + b1 , a1 + b2, , a1 + bq2 - 2, a1+ bq2 - 1, a1+ bq2,

    a1, a2, , aq1 - 2, aq1 - 1, aq1 ) represent a tree code on q1 + q2 edges.

  • Tree Generation Theorems Using

    - valuable tree codes

    2. (a1+ b1, a1+b2, , a1+ bq2 - 2, a1+bq2 - 1, a1+bq2, a1 aq1, a1 aq1 - 1,a1 aq1 - 2, , a1 a2, 0)

    represent a tree code on q1+ q2 edges.

    3. (q1 1 a1 + b1, q1 1 a1+ b2, q1 1 a1+ b3,

    , q1 1 a1+ bq2 - 2, q1 1 a1+ bq2 - 1, q1 1

    a1+ bq2, q1 1 a1, q1 2 a2, , 2 aq1 - 2, 1

    aq1 - 1, 0) represent a tree code on q1+ q2 edges.

  • Tree Generation Theorems U