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City University of New York (CUNY) City University of New York (CUNY)

CUNY Academic Works CUNY Academic Works

Dissertations, Theses, and Capstone Projects CUNY Graduate Center

5-2018

On Some Geometry of Graphs On Some Geometry of Graphs

Zachary S. McGuirk

The Graduate Center, City University of New York

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On Some Geometry of Graphs

by

Zachary S. McGuirk

A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City

University of New York

2018

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ii

2018 c

Zachary S. McGuirk

All Rights Reserved

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iii

This manuscript has been read and accepted by the Graduate Faculty in Mathematics in satisfaction of the dissertation requirement for the degree of

Doctor of Philosophy.

Professor Melvyn Nathanson

Date Chair of Examining Committee

Professor Ara Basmaijian

Date Executive Officer

Professor Melvyn Nathanson Professor J´ ozef Dodziuk

Professor Rados law Wojciechowski Supervisory Committee

The City University of New York

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iv

Abstract

On Some Geometry of Graphs by

Zachary S. McGuirk

Adviser: Professor Dr. Melvyn Nathanson

In this thesis we study the intrinsic geometry of graphs via the constants that appear

in discretized partial differential equations associated to those graphs. By studying

the behavior of a discretized version of Bochner’s inequality for smooth manifolds

at the cone point for a cone over the set of vertices of a graph, a lower bound for the

internal energy of the underlying graph is obtained. This gives a new lower bound

for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the

curvature constant that appears at the cone point and the size of the vertex set for

the underlying graph. For the sake of completeness, the main analysis for cones is

actually done for cones over subsets of the vertex set. We follow this analysis up

by studying which types of functions can achieve equality in the discrete Bochner

inequality, in particular functions which yield the largest possible curvature bound

at the cone point come with a dynamical definition. We are then able to classify the

space of all such functions via spectral graph theory and recast the regularity of a

graph in terms of the dimension of this space of functions.

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Acknowledgments

“There are no atheists in foxholes’ isn’t an argument against atheism, it’s an argument against foxholes.”

-James K. Morrow

I acknowledge everyone who played a role in my graduate school experience. There were many special individuals with whom I interacted. However, in particular I owe an immense debt of gratitude to Dr. Burton Randol, Dr. J´ ozef Dodziuk, Dr.

Melvyn Nathanson, Dr. Byungdo Park, Dr. Sajjad Lakzian, Dr. Enrique Pujals, and Dr. Rados law Wojciechowski.

Under usual circumstances the above should suffice, however nothing about the last six years has been usual and something really must be said about the tireless contributions of Dr. Burton Randol to my academics and my life. During a par- ticularly sad and difficult stretch of my graduate school experience, Dr. Randol took to meeting with me nearly every day just to keep me sane, thinking about math, and moving forward. His wisdom, cool demeanor, and guidance is likely the only reason I survived my time at the CUNY Graduate Center. Even after things started to calm down, he still met with me once or twice a week. For a long while, he was the only person who would talk to me about mathematics. I spent years toiling in obscurity, chasing in the footsteps of Field’s medalist, trying to prove deep conjectures on my own. If not for the encouragement and sustained support of Dr.

Randol, I wouldn’t be here today. Truly, if not for his tireless efforts over the years

v

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vi

I would have never graduated. Dr. Randol, I’m interminably grateful for everything.

I am also very thankful for the help and support provided to me by Dr. Joseph Hirsch and Dr. Linda Keen.

Furthermore, I am thankful for the camaraderie provided by Dr. Jorge Basilio, Dr. Martin Bendersky, Dr. David Edward Bruschi, Bora Ferlengez, Dr. Aron Fis- cher, Dr. Jorge Florez, Dr. Cihan Karabulut, Dr. Asaf Katz, Dr. Seungwon Kim, Alice Kwon, Dr. Ana Turchetti-Maia, and Johannes Niediek.

In conclusion, I should also like to thank my friends, family, girlfriend, and my

undergraduate school’s mathematics department. Thank you all.

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vii

This thesis is dedicated to my father, Jack P. McGuirk.

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Contents

1 Introduction 1

2 Preliminaries 8

3 Cone Construction 15

3.1 Γ-Calculus for Cones . . . . 19 3.2 Γ

c2

for C(G) . . . . 30

4 Graph Energy 32

5 Generalized Harmonic Functions on Finite Graphs 40 5.1 Unit Spherical Averaging Operator . . . . 42 5.2 Unit Ball Averaging Operator . . . . 46 5.3 Explicit Examples . . . . 49

Bibliography 58

viii

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List of Figures

3.1 Example of a partial cone C(X, G) . . . . 16 3.2 Example of a complete cone C(G) . . . . 17 3.3 Metric spheres centered at p . . . . 18

ix

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Chapter 1 Introduction

The relation between Ricci curvature bounds and the analytic and geometric proper- ties of a smooth Riemannian manifold is a well studied subject in geometric analysis (see, for example, [14]). Lower bounds on the Ricci curvature of a manifold can pro- vide eigenvalue estimates and functional inequalities. Thanks to the work of Sturm [27], [28], [26] and Lott-Villani [22], a notion of lower Ricci curvature bounds has been generalized to the setting of metric-measure spaces via a CD(K, N ) curvature- dimension condition due to Bakry and ´ Emery [3]. Bakry and ´ Emery’s original work provided a lower Ricci curvature bound for a measure space via a generalized heat semigroup with some strong regularity conditions for the generator of that semigroup and a curvature-dimension condition realized by what has come to be known as the Γ-Calculus of Bakry and ´ Emery. This curvature-dimension condition is essentially a discretization of Bochner’s inequality for smooth manifolds and the Γ-Calculus of Bakry and ´ Emery is a method of iterated gradients used to study semigroups of linear operators on functions spaces. Refinements of these discretization methods to simply metric spaces, or rather graphs, have yielded a notion of lower Ricci curvature bounds on graphs (see, for example, [4], [9], [8], [15], [16], [20], [21]).

In this thesis, we make use of Bakry and ´ Emery’s Γ-Calculus to study lower Ricci curvature bounds for cones over subsets of vertices. These cones are then used

1

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CHAPTER 1. INTRODUCTION 2

to study local and global properties of the underlying graph and establish a global Poincar´ e inequality. In addition, studying the space of functions which achieved the maximum curvature at the cone point lead to an interesting family of functions with a dynamical nature, which we were able to classify for finite graphs via the spectral properties of the graph Laplacian. So, in chapter 3 we cover the Γ-Calculus for cones over subsets of vertices, in chapter 4 we apply those results to the cone over the full vertex set and derive a Poincar´ e inequality, and lastly we define generalized harmonic functions on graphs and study the space of such functions.

Let’s begin filling in the above with definitions. First, the curvature-dimension condition:

Definition 1.0.1 (Bakry-´ Emery Curvature-Dimension Condition). Suppose K ∈ R and N ∈ (1, ∞]. We say that a graph G = (V, E) satisfies the curvature-dimension conditions, CD(K, N ), if for every x ∈ V and every f ∈ `

2

(V ),

Γ

2

(f )(x) ≥ (∆f )

2

(x)

N + KΓ

1

(f )(x).

Where, for a linear operator L acting on a Hilbert space,

Γ

1

(f, g) = 1

2 [ L(fg) − Lf · g − f · Lg] ,

and

Γ

2

(f, g) = 1

2 [ LΓ

1

(f, g) − Γ

1

( Lf, g) − Γ

1

(f, Lg)]

and in this setting we have taken L to be the unnormalized graph Laplacian ∆ (For more detail, see the beginning of chapter 2).

Also, note that when N = ∞, the second term in the inequality above is under- stood to be 0.

An object of interest in this thesis is the complete cone, C(G), over a finite graph

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CHAPTER 1. INTRODUCTION 3

G = (V, E). In the smooth and metric-measure settings, there is a close relation between the lower Ricci bounds of the space, X, and the lower Ricci curvature bounds of the cones over X. For example, a result due to Bacher and Sturm says that the Ricci curvature of a complete n-dimensional Riemannian manifold M is bounded below by n − 1 if and only if the cone over M (as a metric-measure space) satisfies the CD(0, n + 1) condition [2]. In the setting of CD(K, N ) metric-measure spaces the relation between the weak Ricci curvature bound of X and that of the cone(s) over X has been explored in [17]. We pursue a similar vein in this thesis by studying cones over subsets of the vertices of a graph and looking for relations between the curvature bounds for the cone versus the curvature bounds for the underlying graph.

Formally, C(G) is constructed by taking the graph Cartesian product of G and the complete graph with two vertices K

2

, i.e. GK

2

, where K

2

= ({q, p}, {(q, p)}), and then identifying all the vertices whose second component is p. This complete cone is often referred to as the cone over the vertices of a finite graph. Furthermore, C(X, G) is the subgraph of C(G) restricted to the subset X ⊂ V (G). Note, G needs to be finite so that C(G) is locally-finite and summability concerns can be avoided.

Lastly, in this paper the point p will always refer to the cone point of whatever cone is currently under consideration.

When studying the properties of an underlying graph G that can be extracted

when the cone over G it is convenient to restrict the CD(K, N ) curvature-dimension

conditions to just the cone point (a property which will be called the conical curvature-

dimension, or CCD(K, N ) condition). This interest in the curvature at just the cone

point stems from the fact that the CD(K, N ) inequality depends on vertices which

are at most two steps away, with respect to the graph metric, on the left hand side,

however the right hand side is made up of operators that only depend on vertices that

are at most one away. So when considering a function whose value at the cone point

is zero, the right hand side will only depend on a function’s values on the underlying

graph and it bounds from below the left hand side which will take into account the

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CHAPTER 1. INTRODUCTION 4

effect of adding on the cone point. Furthermore, when taking f (p) = 0, the right hand side of the CD(K, N ) inequality begs for a Cauchy-Schwarz application. Thus, we have:

Definition 1.0.2 (CCD(K, N ) Conical Curvature-Dimension Conditions [19]). Let G = (V, E) be a finite, connected, undirected, loop-edge free graph and consider the cone over the vertex set of G. G is said to satisfy the conical curvature-dimension condition, CCD(K, N ) for K ∈ R and N ∈ (1, ∞], if the cone over G satisfies the CD(K, N ) curvature-dimension conditions at the vertex p, namely if

Γ

c2

(f )(p) ≥ ∆

c

f 

2

(p)

N + KΓ

c1

(f )(p),

holds for any function f defined on the cone and ∆

c

, Γ

c1

and Γ

c2

are the usual ∆, Γ

1

and Γ

2

operators (see (2.4), (2.7) and (2.8)) except on the cone C(G) over G. We note that the second term in (2.1) is understood to be zero when N = ∞.

Furthermore, we refer to the curvature at the cone point K as the conical curva- ture of the finite graph G.

While studying the functions that result in the largest possible curvature at the cone point over a set of vertices, it became natural to consider functions that evenly spread their `

2

(V, µ) mass over V in the limit of iterated applications of a unit spherical averaging operator on G. Thus we define a unit spherical averaging operator and a unit ball averaging operator.

Definition 1.0.3. Let G = (V, E) be a simple, finite, connected, unweighted, loop- edge free graph and let f : V → R. The unit spherical averaging operator is then given by g : R

|V |

→ R

|V |

, such that:

g[f ](v) = 1 deg(v)

X

u∼v

f (u)

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CHAPTER 1. INTRODUCTION 5

for all v ∈ V .

The unit ball averaging operator is then:

Definition 1.0.4. Let G = (V, E) be a simple, finite, connected, unweighted, loop- edge free graph and let f : V → R. The unit ball averaging operator is then given by h : R

|V |

→ R

|V |

, such that:

h[f ](v) = 1

deg

G

(v) + 1 f (v) + X

u∼v

f (u)

!

for all v ∈ V .

For completeness, we note here that by the average of f : V → R, avg(f ), we mean

|V |1

P

v∈V

f (v). Now, We have made these definitions in order to say the following:

Definition 1.0.5. Let G = (V, E) be a simple, finite, connected, unweighted, loop- edge free graph and let f : V → R. We call f generalized harmonic of type I if and only if,

i→∞

lim g

i

[f ](v) = avg(f )

for all v ∈ V and we call f generalized harmonic of type II if and only if,

i→∞

lim h

i

[f ](v) = avg(f )

for all v ∈ V .

With this definition, g

i

[f ](v) = g[g

i−1

[f ]](v), g

0

[f ](v) = f (v) and g

1

[f ](v) = g[f ](v).

And, similarly for h, h

i

[f ](v) = h[h

i−1

[f ]](v), h

0

[f ](v) = f (v), and h

1

[f ](v) = h[f ](v).

Now we can state our main theorems and corollaries:

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CHAPTER 1. INTRODUCTION 6

Theorem 1.0.6. If a finite graph G satisfies CCD(K, N ) curvature-dimension con- dition, then for any function f on V (G) one has

X

y∈V

Γ

1

(f )(y) ≥ 2 − N 2N

 X

y∈V

f (y)



2

+ 2K + |V | − 3 4

X

y∈V

f

2

(y).

For functions f with avg(f ) = 0, this reduces to the following global Poincar´ e in- equality,

kf k

2

≤ s

4

2K + |V | − 3 k∇f k

2

,

where k∇f k

2

is understood in the graph setting to be 2 · P

y∈V

Γ

1

(f )(y).

Theorem 1.0.7. For any finite graph G and a given N > 1 the conical curvature cannot exceed the following number:

K

maxc

= |V | 2 + 3

2 − 2 |V | N .

Theorem 1.0.8 (Curvature Maximizers). Suppose a finite graph G satisfies CCD(K

maxc

, N ).

Then any function f realizes K

maxc

if and only if f is either constant or f − avg(f ) is an eigenfunction corresponding to λ

1

(G) =

N −24N

|V |.

Furthermore, when G is a complete graph, f must be constant (harmonic).

Corollary 1.0.9 (Ricci Curvature of Complete Graphs). Suppose G is the complete graph on n vertices, then the CD(K, N ) property coincides with the CCD(K

c

, N ) condition on the complete subgraph with n − 1 vertices and the curvature of G is

n

2

+ 1 − 2

(n−1)N

.

Furthermore any function that realizes this curvature bound is constant (har-

monic).

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CHAPTER 1. INTRODUCTION 7

Remark 1.0.10. When N = ∞, our bound K

maxc

= 1 +

n2

coincides with the maximum Ricci curvature of complete graphs as found in [18].

The following theorem illustrates an applications of our Γ-calculus on cones:

Theorem 1.0.11. Suppose a finite graph G satisfies CD(K, ∞) for K ≤

12

then the subgraph G ⊂ C(G) satisfies CD(K +

12

, ∞).

Theorem 1.0.12. The set of generalized harmonic functions on a finite graph forms a subspace of `

2

(V, µ) that contains harmonic functions.

Theorem 1.0.13. A function on the vertices of a connected, finite, undirected, non- bipartite graph is generalized harmonic of type I if and only if its weighted average is equal to its average, i.e. if and only if

P

v∈V

deg(v)f (v) P

v∈V

deg(v) = 1

|V | X

v∈V

f (v).

Theorem 1.0.14. Given a connected, undirected, non-bipartite, finite graph G = (V, E) and f : V (G) → R, f is generalized harmonic of type II if and only if for H = GK

2

, with V (H) = {(v, t) ∈ V (G) × {0, 1} and F : V (H) → R such that F (v, t) = f (v), lim

i→∞

g

i

[F ](x) = avg(F ).

The next result has to do with the behavior of δ

v

(for v ∈ V ) functions on the complete graph with m vertices. It provides an explicit formula for the n-th level iteration of g[δ

v

] on G.

Theorem 1.0.15. For any y ∼ x ∈ K

m

, and any n,

g

n

x

](y) =

 

 

1 (m−1)n

P

n2−1

j=0

n 2+j

2j+1

(m − 1)

n2−1−j

(m − 2)

2j+1

; n ≡ 0 (mod 2)

1 (m−1)n

P

n−12

j=0

n−1 2 +j

2j

(m − 1)

n−12 −j

(m − 2)

2j

; n ≡ 1 (mod 2)

.

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Chapter 2

Preliminaries

The fundamental objects in this thesis are graphs, denoted G = (V, E). Unless otherwise stated these graphs will be connected, simple, unweighted, finite, and loop-edge free.

Let L be a linear operator on the Hilbert space of square-summable functions from the set of vertices, V , to the real numbers with measure µ and inner product hf, gi := P

v∈V

µ(v)f (v)g(v); i.e. L acts on

`

2

(V, µ) = {f : V → R : X

v∈V

µ(v)|f (v)|

2

< ∞}. (2.1)

The Γ-Calculus of Bakry and ´ Emery associated to L is then defined (see [3]) as:

Γ

1

(f, g) = 1

2 [ L(fg) − Lf · g − f · Lg] , (2.2)

and

Γ

2

(f, g) = 1

2 [ LΓ

1

(f, g) − Γ

1

( Lf, g) − Γ

1

(f, Lg)] . (2.3)

Given any graph, G, there is an associated adjacency operator, A(G), for which the A

i,j

-entry is 1 whenever there is an edge from vertex i to vertex j and 0 otherwise.

8

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CHAPTER 2. PRELIMINARIES 9

Since we have taken G to be finite, A(G) will be a finite dimensional, square ma- trix. This allows us to avoid summability issues. It should be noted that when G is undirected, A(G) is symmetric.

In this thesis the linear operator that we will be studying is the combinatorial graph Laplacian (see [11]), both unnormalized (denoted ∆) and normalized (denoted

∆) as given by: ¯

∆f (v) = X

u∈V ∼v

(f (u) − f (v)) (2.4)

and

∆f (v) = ¯ 1 deg(v)

X

u∈V ∼v

(f (u) − f (v)), (2.5)

where u ∈ V ∼ v (if the that contains u is unambiguous or the entire vertex set then we simply write u ∼ v) means that u ∈ V and there exists an edge (u, v) ∈ E, and

deg(v) = #{u : u ∼ v}.

Note: When using the unnormalized operator, ∆, the measure µ in `

2

(V, µ) will be the counting measure, i.e. µ(v) = 1, for all v ∈ V . However, a standard modification to the Hilbert space for the normalized operator, ¯ ∆, results in µ(v) = deg(v) for all v ∈ V (see [12]). Let D(G) be the diagonal degree operator for a graph G. Thus,

D(G) = (d

i,j

) =

 

 

deg(i) , when i = j 0 , otherwise

.

The combinatorial graph Laplacians can then be realized as the difference be-

tween the operators D(G) and A(G), i.e. −∆ = (D(G) − A(G)) and − ¯ ∆ =

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CHAPTER 2. PRELIMINARIES 10



I − D

12

AD

12



(see [6]). A graph is said to be k-regular if the degree of every vertex is k. When G is k-regular, −∆ = (k · I − A) or − ¯ ∆ = I −

k1

· A (see [10]).

When thinking of the graph Laplacians (both normalized and unnormalized) as op- erators on the function space of a finite graph, they are realized to be matrices with real entries. For undirected graphs these matrices will be symmetric and therefore self-adjoint. It should be noted that after the work of Wojciechowski (see [29]), these self-adjoint matrices can be extend uniquely to a self-adjoint linear operator acting on the Hilbert space of functions on an infinite graph’s set of vertices. However, for this thesis we won’t need the essential self-adjointedness of the Laplacian on graphs.

Furthermore, the first non-zero eigenvalue of the graph Laplacian, on a locally-finite graph, may be bounded below via the Rayleigh-Ritz quotient. However in the finite setting, which is the case for this thesis, the Rayleigh-Ritz quotient actually yields the first non-zero eigenvalue.

λ

1

= inf



1 2

P

v∈V

P

u∼v

(f (u) − f (v))

2

P

v∈V

f

2

(v) : f ∈ `

2

(V ) and avg(f ) = 0

 .

Applying the definitions for the Γ-Calculus of Bakry-´ Emery to the combinatorial graph Laplacian, one can see that

Γ

1

(f, g)(v) = 1

2 [∆(f g)(v) − ∆f (v)g(v) − f (v)∆g(v)] (2.6)

= 1 2

X

u∈V ∼v

(f (u) − f (v))(g(u) − g(v)), (2.7)

and so in this discrete setting we think of Γ

1

(f, g) as h∇f, ∇gi.

Furthermore,

Γ

2

(f, g)(v) = 1

2 [∆Γ

1

(f, g)(v) − Γ

1

(∆f, g)(v) − Γ

1

(f, ∆g)(v)] . (2.8)

Remark 2.0.1. It should be noted that both ∆f and ¯ ∆f are invariant under the

addition of a constant to f , i.e. ∆f = ∆(f + c) and ¯ ∆f = ¯ ∆(f + c). As a result

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CHAPTER 2. PRELIMINARIES 11

Γ

1

(f, g) and Γ

2

(f, g) are invariant under translations by constant functions, in both f and g. Furthermore, any scalar c ∈ R applied to f , or in the case of Γ

1

and Γ

2

g as well, may be factored out from the operator’s argument.

Now in the case of the normalized graph Laplacian there is by definition a factor of

deg(v)1

that comes up. Hence,

Γ ¯

1

(f, g)(v) = 1 2deg(v)

X

u∈V ∼v

(f (u) − f (v))(g(u) − g(v))

= 1

deg(v) Γ

1

(f, g)(v),

and

Γ ¯

2

(f, g) = 1

deg

2

(v) Γ

2

(f, g)(v).

Let Γ

1

(f, f ) := Γ

1

(f ). Thus, Γ

1

(f )(v) is thought of as |∇f (v)|

2

in the unnormalized setting and one can verify (provided that ∆ is bounded, which is true for finite graphs) the following divergence identity,

k∇f k

22

= X

v∈V

Γ

1

(f )(v) = − X

v∈V

f (v)∆f (v).

In thinking of Γ

1

(f )(v) as |∇f (v)|

2

one is lead to a discrete notion of energy for a graph G via its Dirichlet sum [6].

Definition 2.0.2. The energy of a finite, simple, undirected, unweighted graph is given by:

k∇f k

22

= 1 2

X

v∈V

X

u∼v

(f (u) − f (v))

2

= X

v∈V

Γ

1

(f )(v).

We can now unambiguously define the Curvature-Dimension condition of Bakry-

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CHAPTER 2. PRELIMINARIES 12

Emery. ´

Definition 2.0.3. If K ∈ R and N ∈ (1, ∞] are such that for every v ∈ V and every f ∈ `

2

(V, µ),

Γ

2

(f )(v) ≥ (∆f )

2

(v)

N + KΓ

1

(f )(v),

then we say that G = (V, E) satisfies the CD(K, N ) curvature-dimension condition.

Note, when N = ∞ the middle term in the inequality above is understood to be 0. The largest possible value of K such that the above inequality holds for all v ∈ V and all f ∈ `

2

(V, µ) is thought of as the lower “Ricci curvature” bound, in analogy (see [13]) to the lower Ricci curvature bound, k, that appears in Bochner’s famous inequality for manifolds:

1

2 ∆|∇f |

2

− h∇f, ∇∆f i ≥ 1

n |∆f |

2

+ k|∇f |

2

,

where f is a smooth function on a manifold M , n is an upper bound for its dimen- sion, and k is a lower bound for the Ricci curvature. Furthermore, the CD(K, N ) condition,

Γ

2

(f )(v) ≥ (∆f )

2

(v)

N + KΓ

1

(f )(v),

is invariant under translations and scalar multiplication of f. i.e. let c ∈ R

Γ

2

(c · f )(v) ≥ (∆c · f )

2

(v)

N + KΓ

1

(c · f )(v)

⇔ c

2

Γ

2

(f )(v) ≥ c

2

(∆f )

2

(v)

N + c

2

1

(f )(v)

⇔ Γ

2

(f )(v) ≥ (∆f )

2

(v)

N + KΓ

1

(f )(v)

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CHAPTER 2. PRELIMINARIES 13

and,

Γ

2

(f + c)(v) ≥ (∆(f + c))

2

(v)

N + KΓ

1

(f + c)(v)

⇔ Γ

2

(f )(v) ≥ (∆f )

2

(v)

N + KΓ

1

(f )(v).

This curvature-dimension condition provides an avenue for defining a notion of Ricci curvature for graphs and we make the following definitions.

Definition 2.0.4 (Uniform and Pointwise Ricci Curvature Bounds). We define the pointwise curvatures by

Ric

N

(y) := sup

f

{K : Γ

2

(f )(y) ≥ 1

N (∆f )

2

(y) + KΓ

1

(f )(y)}

and

Ric

(y) := sup

f

{K : Γ

2

(f )(y) ≥ KΓ

1

(f )(y)},

and similarly we define the dimensional (respectively, dimensionless) Ricci curvature of the graph G, Ric

N

(G) (respectively, Ric

(G)) by,

Ric

N

(G) := sup {K : G satisfies CD(K, N )}

and

Ric

(G) := sup {K : G satisfies CD(K, ∞)} .

Definition 2.0.5 (Conical Ricci Curvatures). We define the conical Ricci curvature by

CRic

N

(G) := sup{K : G satisfies CCD(K, N )},

(24)

CHAPTER 2. PRELIMINARIES 14

where CCD(K, N ) is defined in 2.1 and

CRic

(G) := sup{K : G satisfies CCD(K, ∞)}.

Graphs come with a natural metric, which is simply the sum of the least number of edge one must cross to get from vertex u to vertex v in the graph.

d

G

(u, v) = inf (

m−1

X

i=0

1 : v = x

0

∼ x

1

∼ x

2

∼ . . . ∼ x

m

= u )

.

Given F ⊂ V , then

∂F := {(u, v) ∈ E : u ∈ F and v ∈ V \ F }.

From which we can define the finite graph isoperimetric constant,

h(G) := inf

 |∂F |

min{|F |, |V \ F |} : 0 < |F | < ∞

 .

For convenience, let m = sup{deg(v) : v ∈ V } and recall a well-known result of Dodziuk and Alon-Milman from the 1980’s (see [11], [1], [7]):

Theorem 2.0.6. Let G = (V, E) be a finite, simple, connected, loop-edge free graph and let λ

1

be the first non-trivial eigenvalue of ∆ and let m = max{deg(v) : v ∈ V }.

Then, the following inequalities hold:

λ

1

2 ≤ h(G) ≤ p 2mλ

1

.

Definition 2.0.7. Let {G

i

}

i≥0

be a countably infinite family of graphs with sup{deg(v) ; v ∈ V (G

i

)} = m

i

≤ M for some M ∈ N, ordered so that |V

i

| ≤ |V

i+1

| and lim

i→∞

|V

i

| →

∞. If there exists a constant, C > 0, such that h(G

i

) ≥ C > 0, for all i ≥ 0, then

{G

i

}

i≥0

is an expanding family of graphs (see, [24] and [10]).

(25)

Chapter 3

Cone Construction

The complete cone, C(G), over a finite graph G is constructed by taking the graph Cartesian product of G and H, GK

2

, where K

2

= ({q, p}, {(q, p)}) is the complete graph on two vertices q and p, and then identifying all the vertices whose second component is p. In this thesis, as mentioned before, p always refers to the cone point of C(G). More generally, for a subset X ⊂ V (G), the partial cone, C (X, G), is a subgraph of C(G) containing X, all edges (x, y), for x, y ∈ X and all edges (x, p), for x ∈ X. For brevity, we will use a superscript

c

to denote any operation that is taking place on a cone over G or on a subset of the vertices X ⊂ V (G). Notice that any vertex v ∈ V (G) can be thought of as the cone point of the 1-sphere based at v, i.e. S

v1

:= {y ∈ V | y ∼ v} = X in the above construction. In this way partial cones can be useful in studying cliques. The Figures 3.1 and 3.2 on the next page demonstrate examples of a partial cone and, when X = V , the complete cone over the vertices of a finite graph respectively.

First we will prove a few lemmas that calculate the ∆ and Γ operators of a partial cone in terms of the analogous operators on the base graph plus some error terms. In this way the equations and lemmas that follow are pointwise equivalences between operators acting on a cone and operators acting on an underlying graph.

The last subsection is then devoted to an immediate result. In the lemmas that

15

(26)

CHAPTER 3. CONE CONSTRUCTION 16

follow a helpful visual is that of a tower stratifying the metric spheres in C(G) that are centered at the cone point p (see Figure 3.3). Underlying the usefulness of this tower is the fact that the operator Γ

2

(f )(v) depends on vertices that are at most two steps away from v, while for Γ

1

(f )(v) and ∆f (v) only depend on vertices that are one step away.

Figure 3.1: Example of a partial cone C(X, G)

(27)

CHAPTER 3. CONE CONSTRUCTION 17

Figure 3.2: Example of a complete cone C(G)

(28)

CHAPTER 3. CONE CONSTRUCTION 18

Figure 3.3: Metric spheres centered at p

(29)

CHAPTER 3. CONE CONSTRUCTION 19

3.1 Γ-Calculus for Cones

In this section we compute the operators ∆

c

, Γ

c1

, and Γ

c2

in terms of ∆, Γ

1

, and Γ

2

, plus some error terms that result from the addition of the point p to G, for a cone C(X, G) over a finite graph G = (V, E) with X ⊂ V some arbitrary subset of G’s vertices. For convenience we take C = V ∪ {p} to be the vertex set of C(X, G).

Since the operators ∆ and Γ are invariant under translation by a constant, we may assume, without loss of generality, that f (p) = 0.

Denote by S

pn

and B

pn

the metric spheres and balls (resp.) with radius n and center p in the cone. For any subset B ⊂ V , the notation v ∈ B ∼ x means v ∈ B and v ∼ x. Thus, if x ∈ X ⊂ V , then x ∈ C ∼ p and in particular we often say x ∈ S

p1

⊂ V .

Remark 3.1.1. Note that ∆ and Γ

1

only depend on vertices that are at most one step away. Thus, ∆

c

f (x) = ∆f (x) and Γ

c1

(f )(x) = Γ

1

(f )(x) when x  p.

Lemma 3.1.2. Let f be a function on the cone C(X, G) with f (p) = 0 then,

c

f (x) =

 

 

∆f (x) − f (x), x ∼ p P

y∈Sp1

f (y), x = p .

Proof. 1. If x ∼ p, then

c

f (x) = X

y∈C∼x

f (y) − f (x) 

= X

y∈V ∼x

f (y) − f (x) + f (p) − f (x)

= ∆f (x) − f (x).

(30)

CHAPTER 3. CONE CONSTRUCTION 20

2. If x = p, then

c

f (p) = X

y∈C∼p

f (y) − f (p) 

= X

y∈S1p

f (y).

Lemma 3.1.3. Let f be a function on the cone C(X, G) with f (p) = 0 then,

Γ

c1

(f )(x) =

 

 

Γ

1

(f )(x) +

12

f

2

(x), x ∼ p

1 2

P

y∈S1p

f

2

(y), x = p .

Proof. 1. If x ∼ p, then using (2.7)

Γ

c1

(f )(x) = 1 2

X

y∈C∼x

f (y) − f (x) 

2

= 1 2

X

y∈V ∼x

f (y) − f (x) 

2

+ 1

2 f (p) − f (x) 

2

1

(f )(x) + 1 2 f

2

(x).

2. If x = p, then using (2.7)

Γ

c1

(f )(p) = 1 2

X

y∈V

f (y) − f (p) 

2

= 1 2

X

y∈S1p

f

2

(y).

In the next few lemmas we calculate the constituent parts that appear in the definition of Γ

c2

.

Remark 3.1.4. Note that Γ

c2

depends on vertices at most two steps away. Thus

(31)

CHAPTER 3. CONE CONSTRUCTION 21

Γ

c2

(f )(x) coincides with Γ

2

(f )(x) pointwise when x ∈ V \ B

p2

.

Lemma 3.1.5. Let f be a function defined on the cone C(X, G), and suppose f (p) = 0, then whenever

1. x ∈ S

p2

,

Γ

c1

f, ∆

c

f(x) =Γ

1

f, ∆f (x) − 1 2

X

y∈Sp1∼x

f (y) f (y) − f (x).

2. x ∈ S

p1

,

Γ

c1

f, ∆

c

f (x) =Γ

1

f, ∆f (x) − 1 2

X

y∈Sp1∼x

f (y) − f (x) 

2

− 1

2 f (x) X

y∈S1p

f (y) + 1

2 f (x) X

y∈Sp2∼x

f (y) − f (x) 

+ 1

2 f (x)∆f (x) − 1 2 f

2

(x).

3. x = p,

Γ

c1

f, ∆

c

f(x) = 1 2

X

y∈Sp1

f (y)∆f (y) − 1 2

X

y∈S1p

f

2

(y) − 1 2 ( X

y∈Sp1

f (y))

2

.

(32)

CHAPTER 3. CONE CONSTRUCTION 22

Proof. 1. If x ∈ S

p2

, then using (2.7)

Γ

c1

f, ∆

c

f(x) = 1 2

X

y∈C∼x

f (y) − f (x) 

c

f (y) − ∆

c

f (x) 

= 1 2

X

y∈V \S1p∼x

f (y) − f (x) 

c

f (y) − ∆

c

f (x) 

+ 1 2

X

y∈S1p∼x

f (y) − f (x) 

c

f (y) − ∆

c

f (x) 

= 1 2

X

y∈V \S1p∼x

f (y) − f (x) 

∆f (y) − ∆f (x) 

+ 1 2

X

y∈S1p∼x

f (y) − f (x) 

∆f (y) − f (y) − ∆f (x) 

= 1 2

X

y∈V ∼x

f (y) − f (x) 

∆f (y) − ∆f (x) 

− 1 2

X

y∈Sp1∼x

f (y)(f (y) − f (x))

1

f, ∆f (x) − 1 2

X

y∈S1p∼x

f (y) f (y) − f (x).

(33)

CHAPTER 3. CONE CONSTRUCTION 23

2. If x ∈ S

p1

, then using (2.7)

Γ

c1

f, ∆

c

f(x) = 1 2

X

y∈C∼x

f (y) − f (x) 

c

f (y) − ∆

c

f (x) 

= 1 2

X

y∈Sp2∼x

f (y) − f (x) 

∆f (y) − ∆f (x) + f (x) 

+ 1 2

X

y∈S1p∼x

f (y) − f (x) 

∆f (y) − f (y) − ∆f (x) + f (x) 

+ 1

2 f (p) − f (x)  X

y∈S1p

f (y) − ∆f (x) + f (x) 

= 1 2

X

y∈Sp2∼x

f (y) − f (x) 

∆f (y) − ∆f (x) 

+ 1

2 f (x) X

y∈Sp2∼x

f (y) − f (x) 

+ 1 2

X

y∈S1p∼x

f (y) − f (x) 

∆f (y) − ∆f (x) 

− 1 2

X

y∈Sp1∼x

f (y) − f (x) 

2

− 1

2 f (x) X

y∈Sp1

f (y)

+ 1

2 f (x)∆f (x) − 1 2 f

2

(x)

= 1 2

X

y∈V ∼x

f (y) − f (x) 

∆f (y) − ∆f (x) 

− 1 2

X

y∈Sp1∼x

f (y) − f (x) 

2

− 1

2 f (x) X

y∈Sp1

f (y)

+ 1

2 f (x) X

y∈Sp2∼x

f (y) − f (x) + 1

2 f (x)∆f (x) − 1 2 f

2

(x)

1

f, ∆f (x) − 1 2

X

y∈Sp1∼x

f (y) − f (x) 

2

− 1

2 f (x) X

y∈Sp1

f (y)

+ 1

2 f (x) X

y∈Sp2∼x

f (y) − f (x) + 1

2 f (x)∆f (x) − 1

2 f

2

(x).

(34)

CHAPTER 3. CONE CONSTRUCTION 24

3. If x = p, then using (2.7)

Γ

c1

f, ∆

c

f (p) = 1 2

X

y∈C∼p

f (y) − f (p) 

c

f (y) − ∆

c

f (p) 

= 1 2

X

y∈Sp1

f (y) − f (p) 

∆f (y) − f (y) − X

z∈Sp1

f (z) 

= 1 2

X

y∈Sp1



f (y)∆f (y) − f

2

(y) − f (y) X

z∈Sp1

f (z)



= 1 2

X

y∈Sp1

f (y)∆f (y) − 1 2

X

y∈S1p

f

2

(y) − 1 2

X

y∈Sp1

f (y) 

2

.

Lemma 3.1.6. Let f be a function defined on the cone C(X, G), and suppose f (p) = 0, then whenever

1. x ∈ S

p2

,

c

Γ

c1

(f )(x) =∆Γ

1

(f )(x) + 1 2

X

y∈Sp1∼x

f

2

(y).

2. x ∈ S

p1

,

c

Γ

c1

(f )(x) =∆Γ

1

(f )(x) − Γ

1

(f )(x) + 1

2

 X

y∈S1p∼x

f

2

(y) + X

y∈S1p

f

2

(y)



− deg

G

(x) + 1 2 f

2

(x).

3. x = p,

c

Γ

c1

(f )(x) = X

y∈Sp1

Γ

1

(f )(y) − |S

p1

| − 1 2

X

y∈Sp1

f

2

(y).

(35)

CHAPTER 3. CONE CONSTRUCTION 25

Proof. 1. If x ∈ S

p2

, then

c

Γ

c1

(f )(x) = X

y∈C∼x



Γ

c1

(f )(y) − Γ

c1

(f )(x)



= X

y∈V \Sp1∼x



Γ

c1

(f )(y) − Γ

c1

(f )(x)



+ X

y∈Sp1∼x



Γ

c1

(f )(y) − Γ

c1

(f )(x)



= X

y∈V \Sp1∼x



Γ

1

(f )(y) − Γ

1

(f )(x)



+ X

y∈Sp1∼x



Γ

1

(f )(y) + 1

2 f

2

(y) − Γ

1

(f )(x)



= X

y∈V ∼x



Γ

1

(f )(y) − Γ

1

(f )(x)

 + 1

2 X

y∈S1p∼x

f

2

(y)

=∆Γ

1

(f )(x) + 1 2

X

y∈Sp1∼x

f

2

(y).

(36)

CHAPTER 3. CONE CONSTRUCTION 26

2. If x ∈ S

p1

, then

c

Γ

c1

(f )(x) = X

y∈C∼x



Γ

c1

(f )(y) − Γ

c1

(f )(x)



= X

y∈Sp2∼x



Γ

1

(f )(y) − Γ

1

(f )(x) − 1 2 f

2

(x)



+ X

y∈S1p∼x



Γ

1

(f )(y) − Γ

1

(f )(x) + 1

2 f

2

(y) − f

2

(x) 



+ Γ

c1

(f )(p) − Γ

1

(f )(x) − 1 2 f

2

(x)

= X

y∈Sp2∼x



Γ

1

(f )(y) − Γ

1

(f )(x)



− 1

2 deg

C

(x)f

2

(x)

+ X

y∈Sp1∼x



Γ

1

(f )(y) − Γ

1

(f )(x)



+ 1 2

X

y∈Sp1∼x

f

2

(y) + 1 2

X

y∈S1p

f

2

(y) − Γ

1

(f )(x)

= X

y∈V ∼x



Γ

1

(f )(y) − Γ

1

(f )(x)



− Γ

1

(f )(x)

+ 1 2

X

y∈Sp1∼x

f

2

(y) + 1 2

X

y∈S1p

f

2

(y) − 1

2 (deg

G

(x) + 1)f

2

(x)

=∆Γ

1

(f )(x) − Γ

1

(f )(x) + 1 2

 X

y∈Sp1∼x

f

2

(y) + X

y∈Sp1

f

2

(y)



− deg

G

(x) + 1

2 f

2

(x).

(37)

CHAPTER 3. CONE CONSTRUCTION 27

3. If x = p, then

c

Γ

c1

(f )(p) = X

y∈S1p



Γ

c1

(f )(y) − Γ

c1

(f )(p)



= X

y∈S1p



Γ

1

(f )(y) + 1

2 f

2

(y) − 1 2

X

z∈S1p

f

2

(z)



= X

y∈S1p

Γ

1

(f )(y) + 1 2

X

y∈S1p

f

2

(y) − |S

p1

| 2

X

z∈Sp1

f

2

(z)

= X

y∈S1p

Γ

1

(f )(y) − (|S

p1

| − 1) 2

X

y∈Sp1

f

2

(y).

Lemma 3.1.7. Let f be a function defined on the cone C(X, G), and suppose f (p) = 0, then whenever

1. x ∈ S

p2

Γ

c2

(f )(x) =Γ

2

(f )(x) + 3 4

X

y∈S1p∼x

f

2

(y) − 1

2 f (x) X

y∈S1p∼x

f (y).

2. x ∈ S

p1

Γ

c2

(f )(x) =Γ

2

(f )(x) − 1

2 Γ

1

(f )(x) + 1 2

X

y∈S1p∼x

f (y) − f (x) 

2

+ 1 4

 X

y∈Sp1∼x

f

2

(y) − deg(x)f

2

(x)



− 1

2 f (x)∆f (x) + 1

4

 X

y∈Sp1

f

2

(y) + f

2

(x)



− 1

2 f (x) X

y∈Sp2∼x

(f (y) − f (x)).

(38)

CHAPTER 3. CONE CONSTRUCTION 28

3. x = p

Γ

c2

(f )(x) = 1 2

X

y∈S1p

Γ

1

(f )(y) − 1 2

X

y∈Sp1

f (y)∆f (y)

− |S

p1

| − 3 4

X

y∈Sp1

f

2

(y) + 1 2

 X

y∈Sp1

f (y)



2

.

Proof. 1. If x ∈ S

p2

, then

Γ

c2

(f )(x) = 1

2 ∆

c

Γ

c1

(f, f )(x) − Γ

c1

f, ∆

c

f (x)

= 1

2 ∆Γ

1

(f )(x) + 1 4

X

y∈S1p∼x

f

2

(y) − Γ

1

f, ∆f (x)

+ 1 2

X

y∈Sp1∼x

f (y) f (y) − f (x) 

2

(f )(x) + 3 4

X

y∈S1p∼x

f

2

(y) − 1

2 f (x) X

y∈S1p∼x

f (y).

(39)

CHAPTER 3. CONE CONSTRUCTION 29

2. If x ∈ S

p1

, then

Γ

c2

(f )(x) = 1

2 ∆

c

Γ

c1

(f )(x) − Γ

c1

f, ∆

c

f (x)

= 1

2 ∆Γ

1

(f )(x) − 1

2 Γ

1

(f )(x) + 1 4

X

y∈Sp1∼x

f

2

(y) + 1 4

X

y∈S1p

f

2

(y)

− deg

G

(x) + 1

4 f

2

(x) − Γ

1

f, ∆f(x) + 1

2 X

y∈Sp1∼x

f (y) − f (x) 

2

− 1

2 f (x) X

y∈S2p∼x

f (y) − f (x) 

+ 1

2 f (x) X

y∈Sp1

f (y) − 1

2 f (x)∆f (x) + 1 2 f

2

(x)

=  1

2 ∆Γ

1

(f )(x) − Γ

1

f, ∆f (x)



− 1

2 Γ

1

(f )(x) + 1

2 X

y∈Sp1∼x

f (y) − f (x) 

2

+ 1 4

 X

y∈S1p∼x

f

2

(y) − deg

G

(x)f

2

(x)



+ 1 4

X

y∈Sp1

f

2

(y) − 1

2 f (x) X

y∈Sp2∼x

(f (y) − f (x)) − 1

2 f (x)∆f (x) + 1

4 f

2

(x)

2

(f )(x) − 1

2 Γ

1

(f )(x) + 1 2

X

y∈Sp1∼x

f (y) − f (x) 

2

+ 1 4

 X

y∈S1p∼x

f

2

(y) − deg

G

(x)f

2

(x)



− 1

2 f (x)∆f (x) + 1

4

 X

y∈S1p

f

2

(y) + f

2

(x)



− 1

2 f (x) X

y∈Sp2∼x

(f (y) − f (x)).

(40)

CHAPTER 3. CONE CONSTRUCTION 30

3. If x = p, then

Γ

c2

(f )(p) = 1

2 ∆

c

Γ

c1

(f )(p) − Γ

c1

f, ∆

c

f (p)

= 1 2

X

y∈Sp1

Γ

1

(f )(y) − |S

p1

| − 1 4

X

y∈Sp1

f

2

(y) − 1 2

X

y∈S1p

f (y)∆f (y)

+ 1 2

X

y∈S1p

f

2

(y) + 1 2

X

y∈Sp1

f (y) 

2

= 1 2

X

y∈Sp1

Γ

1

(f )(y) − 1 2

X

y∈Sp1

f (y)∆f (y) − |S

p1

| − 3 4

X

y∈S1p

f

2

(y)

+ 1 2

X

y∈S1p

f (y) 

2

.

3.2 Γ c 2 for C(G)

When C(X, G) = C(G) is the full cone over V (G), then S

p1

= V and S

p2

= ∅. Thus, Lemma 3.1.7 reduces to the following Lemma.

Lemma 3.2.1.

Γ

c2

(f )(x) =

 

 

Γ

2

(f )(x) +

12

Γ

1

(f )(x) +

14

P

y∈V

f

2

(y) +

14

f

2

(x), x ∼ p P

y∈V

Γ

1

(f )(y) −

|V |−34

P

y∈V

f

2

(y) +

12

P

y∈V

f (y) 

2

, x = p

Proof. Since S

p1

= V and S

p2

= ∅ the first case in Lemma 3.1.7 disappears. In case 2 notice that when S

p1

= V , then

12

P

y∈Sp1∼x

f (y) − f (x) 

2

= Γ

1

(f )(x) and P

y∈S1p∼x

f

2

(y) − f

2

(x) = ∆f

2

(x). Since

12

Γ

1

(f )(x) =

14

∆f

2

(x) −

12

f (x)∆f (x) the case when x ∼ p follows. When x = p, applying the identity (2) gives the desired result.

This leads to the following result regarding the curvature of the cone,

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CHAPTER 3. CONE CONSTRUCTION 31

Theorem 1.0.11. Suppose G satisfies CD(K, ∞) for K ≤

12

then the subgraph G ⊂ C(G) satisfies CD(K +

12

, ∞).

Proof of Theorem 1.0.11. Suppose G satisfies CD(K, ∞) for K ≤

12

. Since G satis- fies CD(K, ∞) then by Lemma 3.2.1 for x ∼ p,

Γ

c2

(f )(x) = Γ

2

(f )(x) + 1

2 Γ

1

(f )(x) + 1 4

X

y∈V

f

2

(y) + 1 4 f

2

(x).

Therefore, given that,

Γ

2

(f )(x) ≥ 1

2 Γ

1

(f )(x).

one has,

Γ

2

(f )(x)+ 1

2 Γ

1

(f )(x) + 1 4

X

y∈V

f

2

(y) + 1 4 f

2

(x)

≥KΓ

c1

(f )(x) − K

2 f

2

(x) + 1

2 Γ

1

(f )(x) − 1

4 f

2

(x) + 1 4

X

y∈V

f

2

(y) + 1 4 f

2

(x)

and so,

Γ

c2

(f )(x) ≥ (K + 1

2 )Γ

1

(f )(x) + 1 4

X

y∈V

f

2

(y) − K 2 f

2

(x).

Since K ≤

12

, then

14

P

y∈V

f

2

(y) −

K2

f

2

(x) ≥ 0. Hence we may drop both terms

from the inequality and C(G) satisfies CD(K +

12

, ∞) for x ∼ p.

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Chapter 4

Graph Energy

Let us recall here some pertinent definitions for readability. First, the Conical Curvature-Dimension condition:

Definition 4.0.1 (CCD(K, N ) Conical Curvature-Dimension Conditions [19]). 1.0.2 Let G = (V, E) be a finite, connected, undirected, loop-edge free graph and consider the cone over the vertex set of G. G is said to satisfy the conical curvature-dimension condition, CCD(K, N ) for K ∈ R and N ∈ (1, ∞], if the cone over G satisfies the CD(K, N ) curvature-dimension conditions at the vertex p, namely if

Γ

c2

(f )(p) ≥ ∆

c

f 

2

(p)

N + KΓ

c1

(f )(p),

holds for any function f defined on the cone and ∆

c

, Γ

c1

and Γ

c2

are the usual ∆, Γ

1

and Γ

2

operators (see (2.4), (2.7) and (2.8)) except on the cone C(G) over G. We note that the second term in (2.1) is understood to be zero when N = ∞.

Furthermore, we refer to the curvature at the cone point K as the conical curva- ture of the finite graph G.

Definition 4.0.2 (Conical Ricci Curvatures). We define the conical Ricci curvature

32

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CHAPTER 4. GRAPH ENERGY 33

by

CRic

N

(G) := sup{K : G satisfies CCD(K, N )},

where CCD(K, N ) is defined in 2.1 and

CRic

(G) := sup{K : G satisfies CCD(K, ∞)}.

Lastly, we recall the definition for the energy of a graph.

Definition 4.0.3. The energy of a finite, simple, undirected, unweighted graph is given by:

k∇f k

22

= 1 2

X

v∈V

X

u∼v

(f (u) − f (v))

2

= X

v∈V

Γ

1

(f )(v).

If the cone C(G), over a finite graph G, satisfies the CD(K, N ) inequality at the vertex, p, then by Lemmas 3.1.3 and 3.1.2, we get

Γ

c2

(f )(p) ≥ 1 N

X

y∈V

f (y) 

2

+ K 2

X

y∈V

f

2

(y) (4.1)

= 1 N

X

y∈V

f (y) 

2

+ K

2 kf k

22

(4.2)

This leads to the following,

Theorem 1.0.6. If a finite graph, G, satisfies the CCD(K, N ) curvature-dimension condition, then for any function f ∈ `

2

(V ) on G one has

X

y∈V

Γ

1

(f )(y) ≥ 2 − N 2N

 X

y∈V

f (y)



2

+ 2K + |V | − 3

4 kf k

22

. (4.3)

For functions f ∈ `

2

(V ) with avg(f ) = 0, this reduces to the following global

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CHAPTER 4. GRAPH ENERGY 34

Poincar´ e inequality,

kf k

2

≤ s

4

2K + |V | − 3 k∇f k

2

,

where k∇f k

2

is the energy of the graph G that the cone is based on.

Proof of Theorem 1.0.6. Suppose a finite graph G satisfies CCD(K, N ) condition.

By Lemma 3.2.1,

Γ

c2

(f )(p) = X

y∈V

Γ

1

(f )(y) − |V | − 3 4

X

y∈V

f

2

(y) + 1 2

X

y∈V

f (y) 

2

, (4.4)

Upon combining (4.4) and (4.2), we arrive at

X

y∈V

Γ

1

(f )(y) − |V | − 3 4

X

y∈V

f

2

(y) + 1 2

X

y∈V

f (y) 

2

≥ 1 N

X

y∈V

f (y) 

2

+ K 2

X

y∈V

f

2

(y),

which simplifies to

X

y∈V

Γ

1

(f )(y) ≥ 2 − N 2N

X

y∈V

f (y) 

2

+ 2K + |V | − 3 4 kf k

22

.

For f ∈ `

2

(V ) with avg(f ) = 0, the above reduces to

X

y∈V

Γ

1

(f )(y) ≥ 2K + |V | − 3 4 kf k

22

.

Thinking of Γ

1

(f ) as a discrete analogue for |∇f |

2

and recalling 4.1 this yields the Poincar´ e inequality,

kf k

2

s 4

2K + |V | − 3 k∇f k

2

, when f ∈ `

2

(V ), and avg(f ) = 0.

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