An Example of a Vertex Replacement Rule
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An Example of a Vertex Replacement Rule that gives Convergence as well as Exponential Growth. Casey Fye Advisor: Dr. Michelle Previte Penn State Erie, The Behrend College April 2009. Graphs. Definition (roughly): A graph consists of two things Points (vertices)

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An Example of a Vertex Replacement Rule that gives Convergence as well as Exponential Growth

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An example of a vertex replacement rule that gives convergence as well as exponential growth

An Example of a Vertex Replacement Rule

that gives Convergence as well as

Exponential Growth

Casey Fye

Advisor: Dr. Michelle Previte

Penn State Erie, The Behrend College

April 2009


Graphs

Graphs

  • Definition (roughly): A graph consists of two things

    • Points (vertices)

    • line segments (called edges)

      • the endpoints are in the set of vertices

  • We will consider only graphs whose edges have the same length

  • vertex

    edge


    Vertex replacement rules

    Vertex Replacement Rules

    • A vertex replacement rule Ris a finite set of finite graphs called replacement graphs, satisfying two conditions.

      • Each replacement graph has a designated set of vertices called boundary vertices

    H1

    H2


    A replacement rule acts on g

    A Replacement Rule Acts on G

    v1

    v1

    H1

    w1

    w2

    w1

    w2

    v3

    v2

    w3

    G

    v2

    v3

    w3

    H2

    R(G)


    Symmetric condition

    Symmetric Condition

    • A vertex replacement rule R is a finite set of finite graphs called replacement graphs

      • Each Hi must be symmetric with respect to its boundary vertices


    Distinct number of boundary vertices condition

    Distinct Number of Boundary Vertices Condition

    • A vertex replacement rule R is a finite set of finite graphs called replacement graphs

      • Each Hi must be symmetric with respect to its boundary vertices

      • Each replacement graph Hi has a distinct number of boundary vertices

    H1

    H2

    G


    R g has replaceable vertices

    R(G) has Replaceable Vertices

    R(G)


    Sequence of replacement graphs

    Sequence of Replacement Graphs

    G

    R(G)

    R2(G)

    R3(G)


    Two possible options for studying r n g

    Two Possible Options for Studying {Rn(G)}

    • Allow Rn(G) to grow as n→∞ and designate a marked point, pn for center of reference in Rn(G)

      • {(G, p0), (R(G), p1), (R2(G), p2), (R3(G), p3), …}


    Nonreplaceable marked point

    Nonreplaceable Marked Point

    (G, p0)

    (R(G), p1)

    (R2(G), p2)

    (R3(G), p3)


    Limit of sequence of marked graphs

    Limit of Sequence of Marked Graphs


    Another example of a sequence of marked graphs

    Another Example of a Sequence of Marked Graphs

    (R(G), p1)

    (G, p0)

    (R2(G), p2 )

    (R3(G), p3 )


    Limit of this sequence of marked graphs

    Limit of this Sequence of Marked Graphs


    Two possible options for studying r n g1

    Two Possible Options for Studying {Rn(G)}

    2.Scale each graph in the sequence to the same size as the initial graph

    • {(G,1), (R(G),1), (R2(G),1), (R3(G),1), …}

    Limit of (Rn(G), 1)

    (G, 1)

    (R(G), 1)

    (R2(G), 1)


    What is the relationship

    What is the Relationship?

    • Limit M of (Rn(G), pn)

      What is its growth?

    • Limit S of (Rn(G), 1)

      What is its dimension?

    What is the relationship between growth of M and dimension of S?


    Growth of the limit of r n g p n

    Growth of the Limit of (Rn(G), pn)

    • The growth function of a locally finite graph G with respect to a vertex p єV(G) is given by

      f(G, p, m) = |{y єV(G): d(p,y) ≤ m, 0 ≤ m}|


    Binary tree

    Binary Tree

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    .

    .

    Growth Function:

    f(G, p, 0)= 20

    f(G, p, 1)= 20+20+21

    f(G, p, 2)= 20+20+21+21+22

    f(G, p, m)= 2m+1+∑2i

    =2m(3)-2

    .

    .

    .

    .

    .

    p

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    .

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    .

    m-1

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    .

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    .

    i=1

    .

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    .

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    .

    .

    .


    Exponential growth of the limit of r n g p n

    Exponential Growth of the Limit of (Rn(G), pn)

    • The growth function of a locally finite graph G with respect to a vertex p єV(G) is given by

      f(G, p, m) = |{y єV(G): d(p,y) ≤ m, 0 ≤ m}|

    • We say G has exponential growth if

      f(G, p, m)≥cam for some constants c>0 and a>1

    • We say G has polynomial growth if

      f(G, p, m) is bounded above by a polynomial


    Theorem for exponential growth j previte and m previte

    Theorem for Exponential Growth(J. Previte and M. Previte)

    • Let (G, p0) be an initial marked graph with at least one replaceable vertex, and let R={H} be a replacement rule such that H has a replaceable vertex but the boundary vertices of H are not replaceable. Suppose that there exist two replaceable vertices v1 and v2 in H and nonreplaceable path in H between v1 and v2 that contains exactly one boundary vertex. Then each graph in the set of limit graphs R∞(G, p∞) of (G, p0) has exponential growth.

    v1

    v2


    What is the relationship1

    What is the Relationship?

    • Limit M of (Rn(G), pn)

      What is its growth?

    • Limit S of (Rn(G), 1)

      What is its dimension?

    What is the relationship between growth of M and dimension of S?


    Theorem for convergence j previte

    Theorem for Convergence(J. Previte)

    • Let H define a vertex replacement rule R and let G be a finite graph with at least one replaceable vertex. Then the normalized sequence {(Rn(G), 1)} converges in the Gromov-Hausdorff metric if and only if one of the following hold:

      • H contains exactly one replaceable vertex

      • |∂H|> 1 and every path in H connecting two different boundary vertices contains at least two replaceable vertices

    Convergence

    Convergence

    Divergence

    H

    H


    What is the relationship2

    What is the Relationship?

    • Limit M of (Rn(G), pn)

      What is its growth?

    • Limit S of (Rn(G), 1)

      What is its dimension?

    • Conjecture:

      M has polynomial growth with degree equal to the dimension of S

    What is the relationship between growth of M and dimension of S?


    Conjecture

    Conjecture

    • M has polynomial growth with degree equal to the dimension of S

      Yes and No

      No.

      My example:

    R


    Exponential growth

    Exponential Growth

    • Let (G, p0) be an initial marked graph with at least one replaceable vertex, and let R={H} be a replacement rule such that H has a replaceable vertex but the boundary vertices of H are not replaceable. Suppose that there exist two replaceable vertices v1 and v2 in H and nonreplaceable path in H between v1 and v2 that contains exactly one boundary vertex. Then each graph in the set of limit graphs R∞(G, p∞) of (G, p0) has exponential growth.

    v1

    v2


    Conjecture1

    Conjecture

    • M has polynomial growth with degree equal to the dimension of S

      Yes

      Dr.’s Joe and Michelle Previte

      proved

      growth of M= dimension of S


    Important application of my example

    Important Application of My Example

    • Definition of Box Dimension≈

      Definition of Hausdorff Dimension

      dimbox(S)=2

      dimhaus(S)=∞


    The sequence r n g

    The Sequence Rn(G)

    R

    R(G)

    R2(G)

    R3(G)


    An example of a vertex replacement rule that gives convergence as well as exponential growth

    Thank you!


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