Costandino Dufort Moraites, Margaret Nichols

Generalized Roundness of Trees
Costandino Dufort Moraites, Margaret Nichols, Luke Levy-Moore
Introduction
Results
Taking a broad view of things, there has been interest in determining when the generalized roundness
of a Cayley graph of a group has positive generalized roundness. This stems from the connection between generalized roundness and the coarse BaumConnes conjecture. In some instances such Cayley
graphs are countable spherically metric trees of generalized roundness one. Finite metric trees, on the other
hand, always have generalized roundness greater than
one. The aim of our project was to examine the still incomplete classification of metric trees of generalized
roundness one. We obtained substantial new classes
of metric trees of generalized roundness one including weighted spherically symmetric trees, asymmetric
weighted infinitely bifurcating trees and certain infinite comb graphs. We also looked at the generalized
roundness of Irrational Trees, a family of countably infinite trees introduced as examples of trees whose endomorphism spaces are Mingo spaces—-a topic that
arises in the study of non-commutative geometry. A
key feature of our work was the isolation of a new
concept in graph theory: asymptotic subgraph isomorphisms.
The Monster
Generalized Roundness
The generalized roundness, q, of a metric space ( X, δ)
is the supremum of the set of all p ≥ 0 which satisfy the following property: For all integers m ≥ 2
and all choices of (not necessarily distinct) points
a1, . . . , am, b1, . . . , bm ∈ X, we have
∑
1≤ i < j ≤ m
{ δ ( a i , a j ) p + δ ( bi , b j ) p } ≤
∑
1≤i,j≤m
δ ( ai , b j ) p
Pictured below is the tree Yn where n is the number
of vertices in an arm of the tree. Understanding the
generalized roundness of a simple graph such as this
one is extremely powerful as it allows us to provide an
upper bound on the generalized roundness of any tree
which contains this tree as a subtree. In general, understanding the generalized roundness of the Yn trees
proved quite difficult, especially for the infinite tree,
Y∞, with an infinite number of nodes in each arm.
diverges and |{ j|d j > 1}| = ℵ0, then
all edges length one. We define the k-comb Gk to be
the finite subgraph of the quintessential comb with k
teeth.
1
1
1
1
1
1
1
1
1
1
1
1
1
Figure 4: The infinite quintessential comb
Figure 2: A SST with downward degree sequence (3,3,2)
A countable rooted metric tree ( T, ρ, v0) is infinitely
bifurcating if v0 has infinitely many descendants with
vertex degree greater than or equal to 3. We call such
an infinitely bifurcating tree divergent if it contains infinitely many radial geodesic paths from v0 with divergent length.
Icicle graphs are similar to comb graphs, except their
gaps and teeth grow in length. Certain icicle graphs
I are ASI to Gk for all k, and thus q( I ) ≤ q( Gk ). As
n of the
1 generalized
2
3
4 roundness
n−1
n +k-comb
1
k → 1∞ the
goes
to one and
so2q( I )3 = 41.
1
n n+1
3
3
4
5
4
6
5
6
Figure 5: An example of an infinite icicle
In general all icicle graphs which have gaps and teeth
growing at the same polynomial rate will be ASI to any
k-comb and hence have generalized roundness one.
Figure 1: The Monster Tree
Below is some interesting data for how the finite Yn
subtrees of Y∞ behave. In order, the columns describe
the number of red vertices (which correspond to the ai
from the generalized roundness inequality) in an arm
of the tree, the number of graphs with that many red
vertices in each arm, and a rough figure of the generalized roundness of this subfamily of graphs.
#
#
red graphs q(Yn )
(data cont.)
1
9
1.532 6
7
1.456
2
5
1.484 7
4
1.449
3
5
1.476 8
11
1.449
4
6
1.463 10
4
1.443
5
4
1.456
Spherically Symmetric Trees
Weighted Spherically Symmetric and
Infinitely Bifurcating Trees
A tree ( T, ρ) is spherically symmetric if we can choose a
root vertex v0 ∈ T such that ρ(v0, v1) = ρ(v0, v2) ⇒
deg(v1) = deg(v2) for all v1, v2 ∈ T. The downward
degree sequence of T is the sequence (dn ) such that a
node at depth k has dk children.
Due to Doust et al. we see that a spherically symmetric
tree with downward degree sequence (dn ) such that
|{ j|d j > 1}| = ℵ0 has generalized roundness one.
A weighted spherically symmetric tree T has an additionally defined downward length sequence (�n ).
Advisor: Anthony Weston, Canisius College
Theorem. If
q( T ) = 1.
∞
∑ i =0 � i
Cornell University
Department of Mathematics
Summer Math Institute
Conclusions
Figure 3: The infinitely bifurcating Fibonacci tree
In particular, if the weight of each edge is within the
interval [�1, �2], where �1 > 0, then T has generalized
roundness one. We note that this condition may be
weakened to encompass a much larger class of such
trees, however, it becomes difficult to describe this
class.
Combs and Icicle Graphs
G�
Loosely speaking a graph
is asymptotically subgraph
isomorphic (ASI) to G if G � contains a subgraph H �
whose edge weights can be normalized to become arbitrarily close to those of G.
Theorem. Let T and T � be weighted trees, and let T � be ASI
to T. Then q( T � ) ≤ q( T ).
This result allows us to use known properties of the
class of comb graphs to determine another collection
of trees has generalized roundness one.
The quintessential comb is an infinite path with a single additional edge (called a tooth) at each node, with
Classification of countable metric trees has proved to
be a difficult task. We found that even the simplest
trees to describe were challenging to analyze, and it
was only once we imposed more structure and symmetry upon the trees that we were able to effectively
compute their generalized roundness.
Acknowledgements
We would like to thank our TAs, Mathav Murugan
and Elizabeth Wesson as well as our advisor Anthony
Weston for helping us throughout our work. In addition we’d like to thank Bruce Hughes for clarifying the
construction of irrational trees.
For Further Information
Costandino Dufort Moraites ([email protected])
Margaret Nichols ([email protected])
Luke Levy-Moore ([email protected])