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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

Casey Fye

Advisor: Dr. Michelle Previte

Penn State Erie, The Behrend College

April 2009

- Definition (roughly): A graph consists of two things
- Points (vertices)
- line segments (called edges)
- the endpoints are in the set of vertices

vertex

edge

- 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

v1

v1

H1

w1

w2

w1

w2

v3

v2

w3

G

v2

v3

w3

H2

R(G)

- 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

- 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)

G

R(G)

R2(G)

R3(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), …}

(G, p0)

(R(G), p1)

(R2(G), p2)

(R3(G), p3)

(R(G), p1)

(G, p0)

(R2(G), p2 )

(R3(G), p3 )

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)

- 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?

- 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}|

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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

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m-1

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i=1

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- 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

- 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

- 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?

- 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

- 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?

- M has polynomial growth with degree equal to the dimension of S
Yes and No

No.

My example:

R

- 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

- 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

- Definition of Box Dimension≈
Definition of Hausdorff Dimension

dimbox(S)=2

dimhaus(S)=∞

R

R(G)

R2(G)

R3(G)

Thank you!