Random Walks On Graphs: Theory & Applications

Random Walks On Graphs:Theory & Applications

Amir Daneshgar

Department of Mathematical Sciences

Sharif University of Technology

August 23, 2004

Random Walks On Graphs: Theory & Applications

Introduction

Basic idea

I G and H are two geometric objects, we are going to compare.

Random Walks On Graphs: Theory & Applications

Introduction

Basic idea

I PG and PH are diffusions on G and H, respectively, related toeach other through the map σ : G −→ H.

Random Walks On Graphs: Theory & Applications

Introduction

Basic idea

I σ has some nice local properties.

e.g. does not decrease local density.

Random Walks On Graphs: Theory & Applications

Introduction

Basic idea

I If these condition are nice enough, we may deduce someproperties of these diffusions that are related to somegeometric properties of the objects.

e.g. PH is faster than PG .

Random Walks On Graphs: Theory & Applications

Introduction

The main setup

I For a geometric object G, the most natural operator relatedto its geometry is the Laplacian.

I Laplacian appears naturally in different diffusion processes.

I Usually the rate of convergence of the deffusion is controlledby the eigenvalues and the eigenfunctions of the Laplacian.

I Eigenfunctions of the natural Laplacian are closely related tothe symmetries of the object.

Random Walks On Graphs: Theory & Applications

Introduction

The main setup

I For a geometric object G, the most natural operator relatedto its geometry is the Laplacian.

I Laplacian appears naturally in different diffusion processes.

I Usually the rate of convergence of the deffusion is controlledby the eigenvalues and the eigenfunctions of the Laplacian.

I Eigenfunctions of the natural Laplacian are closely related tothe symmetries of the object.

Random Walks On Graphs: Theory & Applications

Introduction

The main setup

I For a geometric object G, the most natural operator relatedto its geometry is the Laplacian.

I Laplacian appears naturally in different diffusion processes.

I Usually the rate of convergence of the deffusion is controlledby the eigenvalues and the eigenfunctions of the Laplacian.

I Eigenfunctions of the natural Laplacian are closely related tothe symmetries of the object.

Random Walks On Graphs: Theory & Applications

Introduction

The main setup

I For a geometric object G, the most natural operator relatedto its geometry is the Laplacian.

I Laplacian appears naturally in different diffusion processes.

I Usually the rate of convergence of the deffusion is controlledby the eigenvalues and the eigenfunctions of the Laplacian.

I Eigenfunctions of the natural Laplacian are closely related tothe symmetries of the object.

Random Walks On Graphs: Theory & Applications

Outline

Outline

The main setupNatural random walks on graphsRandom walks and LaplacianSummary of part I

Graph homomorphisms and random walksSome important problems in combinatoricsA couple of resultsSome applications

Epilogue

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

GRAPH: a geometric object

I A graph is a discrete geometric object with a set of vertices(denoting the space) and a set of edges (denoting theconnections).

I In this talk we concentrate on simple graphs.

I But everything can be extended to the general case.

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

GRAPH: a geometric object

I A graph is a discrete geometric object with a set of vertices(denoting the space) and a set of edges (denoting theconnections).

I In this talk we concentrate on simple graphs.

I But everything can be extended to the general case.

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

GRAPH: a geometric object

I A graph is a discrete geometric object with a set of vertices(denoting the space) and a set of edges (denoting theconnections).

I In this talk we concentrate on simple graphs.

I But everything can be extended to the general case.

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Natural random walks on graphs

The natural random walk on a graph

I The natural random walk on a graph G is defined as,

KG(u, v) =

{ 1d

+

G(u)

u → v

0 u 6→ v,

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

The Laplacian

I

K =

0 12 0

12

13 0

13

13

0 12 012

13

13

13 0

∆ = I−K =

1 −12 0 −12

−13 1 −13 −

13

0 −12 1 −12

−13 −13 −

13 1

.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Evolution of a random walk

I If xn is the probability distribution at stage n, then

xn = xn−1KG ⇒ xn = x0KnG.

How can one compute the nth power KnG?

I If the chain is ergodic then the following limit exists and doesnot depend on the initial distribution, x0 ,

limn

x0KnG

= π.

I The stationary distribution π is in the left-kernel of theLaplacian (dually as a measure), i.e.

π = πKG ⇔ π∆ = π(I −KG) = 0.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Evolution of a random walk

I If xn is the probability distribution at stage n, then

xn = xn−1KG ⇒ xn = x0KnG.

How can one compute the nth power KnG?

I If the chain is ergodic then the following limit exists and doesnot depend on the initial distribution, x0 ,

limn

x0KnG

= π.

I The stationary distribution π is in the left-kernel of theLaplacian (dually as a measure), i.e.

π = πKG ⇔ π∆ = π(I −KG) = 0.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Evolution of a random walk

I If xn is the probability distribution at stage n, then

xn = xn−1KG ⇒ xn = x0KnG.

How can one compute the nth power KnG?

I If the chain is ergodic then the following limit exists and doesnot depend on the initial distribution, x0 ,

limn

x0KnG

= π.

I The stationary distribution π is in the left-kernel of theLaplacian (dually as a measure), i.e.

π = πKG ⇔ π∆ = π(I −KG) = 0.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Evolution of a random walk

I If xn is the probability distribution at stage n, then

xn = xn−1KG ⇒ xn = x0KnG.

How can one compute the nth power KnG?

I If the chain is ergodic then the following limit exists and doesnot depend on the initial distribution, x0 ,

limn

x0KnG

= π.

I The stationary distribution π is in the left-kernel of theLaplacian (dually as a measure), i.e.

π = πKG ⇔ π∆ = π(I −KG) = 0.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Fourier transform and diagonalization

I There is a relationship between the solution sets of thefollowing equations,

π = πKG ↔ h = KG h.

I Such a function h is called harmonic.

I If one can diagonalize the kernel KG as KG = PΛP−1, then

in the new coordinates changed by P , everything is simplifiedas,

∆ = I − Λ & (π∆ = 0 ⇔ ∆h = 0).

I How can we find such a diagonalization?

I Fourier transform is available when there are enoughsymmetry in G.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Fourier transform and diagonalization

I There is a relationship between the solution sets of thefollowing equations,

π = πKG ↔ h = KG h.

I Such a function h is called harmonic.

I If one can diagonalize the kernel KG as KG = PΛP−1, then

in the new coordinates changed by P , everything is simplifiedas,

∆ = I − Λ & (π∆ = 0 ⇔ ∆h = 0).

I How can we find such a diagonalization?

I Fourier transform is available when there are enoughsymmetry in G.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Fourier transform and diagonalization

I There is a relationship between the solution sets of thefollowing equations,

π = πKG ↔ h = KG h.

I Such a function h is called harmonic.

I If one can diagonalize the kernel KG as KG = PΛP−1, then

in the new coordinates changed by P , everything is simplifiedas,

∆ = I − Λ & (π∆ = 0 ⇔ ∆h = 0).

I How can we find such a diagonalization?

I Fourier transform is available when there are enoughsymmetry in G.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Fourier transform and diagonalization

I There is a relationship between the solution sets of thefollowing equations,

π = πKG ↔ h = KG h.

I Such a function h is called harmonic.

I If one can diagonalize the kernel KG as KG = PΛP−1, then

in the new coordinates changed by P , everything is simplifiedas,

∆ = I − Λ & (π∆ = 0 ⇔ ∆h = 0).

I How can we find such a diagonalization?

I Fourier transform is available when there are enoughsymmetry in G.

Random Walks On Graphs: Theory & Applications

The main setup

Random walks and Laplacian

Fourier transform and diagonalization

I There is a relationship between the solution sets of thefollowing equations,

π = πKG ↔ h = KG h.

I Such a function h is called harmonic.

I If one can diagonalize the kernel KG as KG = PΛP−1, then

in the new coordinates changed by P , everything is simplifiedas,

∆ = I − Λ & (π∆ = 0 ⇔ ∆h = 0).

I How can we find such a diagonalization?

I Fourier transform is available when there are enoughsymmetry in G.

Random Walks On Graphs: Theory & Applications

The main setup

Summary of part I

Summing up

I Laplacian is an important self-adjoint operator that is closelyrelated to the geometry of its space.

I The eigenvalues of the Laplacian control the rate ofconvergence of some natural diffusions on the space.

I There is a good knowledge about the eigenspace of theLaplacian when the space admits enough symmetry.

I It is possible to compare the rates of convergence of twocoupled diffusions when the coupling is through a nearlycontinuous mapping.

Random Walks On Graphs: Theory & Applications

The main setup

Summary of part I

Summing up

I Laplacian is an important self-adjoint operator that is closelyrelated to the geometry of its space.

I The eigenvalues of the Laplacian control the rate ofconvergence of some natural diffusions on the space.

I There is a good knowledge about the eigenspace of theLaplacian when the space admits enough symmetry.

I It is possible to compare the rates of convergence of twocoupled diffusions when the coupling is through a nearlycontinuous mapping.

Random Walks On Graphs: Theory & Applications

The main setup

Summary of part I

Summing up

I Laplacian is an important self-adjoint operator that is closelyrelated to the geometry of its space.

I The eigenvalues of the Laplacian control the rate ofconvergence of some natural diffusions on the space.

I There is a good knowledge about the eigenspace of theLaplacian when the space admits enough symmetry.

I It is possible to compare the rates of convergence of twocoupled diffusions when the coupling is through a nearlycontinuous mapping.

Random Walks On Graphs: Theory & Applications

The main setup

Summary of part I

Summing up

I Laplacian is an important self-adjoint operator that is closelyrelated to the geometry of its space.

I The eigenvalues of the Laplacian control the rate ofconvergence of some natural diffusions on the space.

I There is a good knowledge about the eigenspace of theLaplacian when the space admits enough symmetry.

I It is possible to compare the rates of convergence of twocoupled diffusions when the coupling is through a nearlycontinuous mapping.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A graph homomorphism

I A graph homomorphism σ from a graph G to a graph H is amap σ : V (G) −→ V (H) such that uv ∈ E(G) impliesσ(u)σ(v) ∈ E(H).

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A graph homomorphism

I A graph homomorphism σ from a graph G to a graph H is amap σ : V (G) −→ V (H) such that uv ∈ E(G) impliesσ(u)σ(v) ∈ E(H).

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A question

I Does there exist a homomorphism from the Petersen graph tothe triangle K3?

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A question

I Does there exist a homomorphism from the Petersen graph tothe triangle K3?

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Graph colouring

I Homomorphisms to Kn is equivalent to colouring the verticesof the graph by n colours such that the terminal ends of eachedge have different colours.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Graph colouring

I Homomorphisms to Kn is equivalent to colouring the verticesof the graph by n colours such that the terminal ends of eachedge have different colours.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Another question!

I Does there exist a homomorphism from the Petersen graph tothe 5-cycle C5?

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Another question!

I Does there exist a homomorphism from the Petersen graph tothe 5-cycle C5?

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.

I The circular colouring problem.I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.

I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.I The fractional colouring problem.

I The graph partitioning problem, specially, existence results indesign theory.

I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.

I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Graph homomorphisms and combinatorics

I Graph homomorphisms are natural maps in the category ofgraphs.

I Many different concepts in combinatorics are related to thehomomorphism problem, e.g.

I The ordinary colouring problem.I The circular colouring problem.I The fractional colouring problem.I The graph partitioning problem, specially, existence results in

design theory.I The Hamiltonicity problem.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some important problems in combinatorics

Algorithmic considerations

I The following problem is NP-complete(P. Hell & J. Nesetril 1990).

Problem: HCOL.Constant: A graph H.

Given: A graph G.Question: Does there exist a homomorphism σ : G −→ H?

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

Some definitions and notations I

I Notations Hom(G, H), Homv(G, H) and Home(G, H)denote the sets of ordinary, onto-vertices and onto-edgeshomomorphisms from G to H, respectively.

I Inverse image parameters: If σ ∈ Hom(G, H) andX, Y ⊆ V (G) we define,

I E(X, Y ) , {uv ∈ E(G) | u ∈ X & v ∈ Y },I Mσ , min

x,y∈V (H){|E(σ−1(x), σ−1(y))| | E(σ−1(x), σ−1(y)) 6= ∅},

I Mσ

, maxx,y∈V (H)

|E(σ−1(x), σ−1(y))|,

I Sσ , minx∈V (H)

|σ−1(x)|,

I Sσ

, maxx∈V (H)

|σ−1(x)|.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

Some definitions and notations I

I Notations Hom(G, H), Homv(G, H) and Home(G, H)denote the sets of ordinary, onto-vertices and onto-edgeshomomorphisms from G to H, respectively.

I Inverse image parameters: If σ ∈ Hom(G, H) andX, Y ⊆ V (G) we define,

I E(X, Y ) , {uv ∈ E(G) | u ∈ X & v ∈ Y },I Mσ , min

x,y∈V (H){|E(σ−1(x), σ−1(y))| | E(σ−1(x), σ−1(y)) 6= ∅},

I Mσ

, maxx,y∈V (H)

|E(σ−1(x), σ−1(y))|,

I Sσ , minx∈V (H)

|σ−1(x)|,

I Sσ

, maxx∈V (H)

|σ−1(x)|.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

Some definitions and notations II

I For a graph G we define AG to be the adjacency matrix of Gand we let DG be the diagonal matrix whose diagonal entrycorresponding to the vertex v, is dv the degree of v. Then thecombinatorial Laplacian of G is defined as ∆G , DG −AG .

I We order the eigenvalues of ∆G as follows,

λG

1≤ λG

2≤ · · · ≤ λG

n,

I We consider the following random walk,

K(u, v) =

1∆ u ∼ v & u 6= v

∆−du∆ u = v

0 otherwise.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

Some definitions and notations II

I For a graph G we define AG to be the adjacency matrix of Gand we let DG be the diagonal matrix whose diagonal entrycorresponding to the vertex v, is dv the degree of v. Then thecombinatorial Laplacian of G is defined as ∆G , DG −AG .

I We order the eigenvalues of ∆G as follows,

λG

1≤ λG

2≤ · · · ≤ λG

n,

I We consider the following random walk,

K(u, v) =

1∆ u ∼ v & u 6= v

∆−du∆ u = v

0 otherwise.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

Some definitions and notations II

I For a graph G we define AG to be the adjacency matrix of Gand we let DG be the diagonal matrix whose diagonal entrycorresponding to the vertex v, is dv the degree of v. Then thecombinatorial Laplacian of G is defined as ∆G , DG −AG .

I We order the eigenvalues of ∆G as follows,

λG

1≤ λG

2≤ · · · ≤ λG

n,

I We consider the following random walk,

K(u, v) =

1∆ u ∼ v & u 6= v

∆−du∆ u = v

0 otherwise.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

A couple of results

A comparison theorem(A. Daneshgar & H. Hajiabolhassan 2002)

I Let G and H be two graphs with |V (G)| = n and|V (H)| = m.a ) If σ ∈ Homv(G, H), then for all 1 ≤ k ≤ m,

λG

k≤ M

σ

Sσλ

H

k.

b ) If σ ∈ Home(G, H), then for all 1 ≤ k ≤ m,

λG

n−m+k≥ MσSσ

λH

k.

c ) If σ ∈ Hom(G, H) and H is both vertex and edge transitivethen,

λG

n≥ 2|E(G)|

n∆H

λH

m.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

hamiltonicity of the Petersen graph!

I If a graph G with n vertices is Hamiltonian then there exists ahomomorphism such as σ ∈ Homv(Cn , G) such that

Sσ = Sσ = Mσ

= 1

.

I Considering λP

5and Part (a) show that Homv(C10 , P ) = ∅.

I Hence, Petersen graph is not Hamiltonian.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

hamiltonicity of the Petersen graph!

I If a graph G with n vertices is Hamiltonian then there exists ahomomorphism such as σ ∈ Homv(Cn , G) such that

Sσ = Sσ = Mσ

= 1

.

I Considering λP

5and Part (a) show that Homv(C10 , P ) = ∅.

I Hence, Petersen graph is not Hamiltonian.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

hamiltonicity of the Petersen graph!

I If a graph G with n vertices is Hamiltonian then there exists ahomomorphism such as σ ∈ Homv(Cn , G) such that

Sσ = Sσ = Mσ

= 1

.

I Considering λP

5and Part (a) show that Homv(C10 , P ) = ∅.

I Hence, Petersen graph is not Hamiltonian.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

hamiltonicity of the Petersen graph!

I If a graph G with n vertices is Hamiltonian then there exists ahomomorphism such as σ ∈ Homv(Cn , G) such that

Sσ = Sσ = Mσ

= 1

.

I Considering λP

5and Part (a) show that Homv(C10 , P ) = ∅.

I Hence, Petersen graph is not Hamiltonian.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Fisher’s inequality!

I Let λH denote the graph on V (H) such that each edge of Hhas multiplicity λ. It is well-known that a 2− (v, k, λ) designon the set V = {1, 2, . . . , v} with b blocks can be consideredas a decomposition of λKv by b copies of Kk .

I In other words, this is equivalent to considering the existenceof a homomorphism σ ∈ Home(∪bi=1Kk ,Kv) for whichMσ = M

σ= λ. It is easy to see that for such a

homomorphism one has r = Sσ = Sσ

= bkv .I To prove Fisher’s inequality, assume that b < v. Then by Part

(a) for the bth eigenvalue we should have k ≤ λr v. But sincer(k − 1) = λ(v − 1) holds for any 2-design, we should haveλ ≥ r and consequently, v = k.

I Hence, v > k ⇒ b ≥ v.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Fisher’s inequality!

I Let λH denote the graph on V (H) such that each edge of Hhas multiplicity λ. It is well-known that a 2− (v, k, λ) designon the set V = {1, 2, . . . , v} with b blocks can be consideredas a decomposition of λKv by b copies of Kk .

I In other words, this is equivalent to considering the existenceof a homomorphism σ ∈ Home(∪bi=1Kk ,Kv) for whichMσ = M

σ= λ. It is easy to see that for such a

homomorphism one has r = Sσ = Sσ

= bkv .I To prove Fisher’s inequality, assume that b < v. Then by Part

(a) for the bth eigenvalue we should have k ≤ λr v. But sincer(k − 1) = λ(v − 1) holds for any 2-design, we should haveλ ≥ r and consequently, v = k.

I Hence, v > k ⇒ b ≥ v.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Fisher’s inequality!

I Let λH denote the graph on V (H) such that each edge of Hhas multiplicity λ. It is well-known that a 2− (v, k, λ) designon the set V = {1, 2, . . . , v} with b blocks can be consideredas a decomposition of λKv by b copies of Kk .

I In other words, this is equivalent to considering the existenceof a homomorphism σ ∈ Home(∪bi=1Kk ,Kv) for whichMσ = M

σ= λ. It is easy to see that for such a

homomorphism one has r = Sσ = Sσ

= bkv .

I To prove Fisher’s inequality, assume that b < v. Then by Part(a) for the bth eigenvalue we should have k ≤ λr v. But sincer(k − 1) = λ(v − 1) holds for any 2-design, we should haveλ ≥ r and consequently, v = k.

I Hence, v > k ⇒ b ≥ v.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Fisher’s inequality!

I Let λH denote the graph on V (H) such that each edge of Hhas multiplicity λ. It is well-known that a 2− (v, k, λ) designon the set V = {1, 2, . . . , v} with b blocks can be consideredas a decomposition of λKv by b copies of Kk .

I In other words, this is equivalent to considering the existenceof a homomorphism σ ∈ Home(∪bi=1Kk ,Kv) for whichMσ = M

σ= λ. It is easy to see that for such a

homomorphism one has r = Sσ = Sσ

= bkv .I To prove Fisher’s inequality, assume that b < v. Then by Part

(a) for the bth eigenvalue we should have k ≤ λr v. But sincer(k − 1) = λ(v − 1) holds for any 2-design, we should haveλ ≥ r and consequently, v = k.

I Hence, v > k ⇒ b ≥ v.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Fisher’s inequality!

I Let λH denote the graph on V (H) such that each edge of Hhas multiplicity λ. It is well-known that a 2− (v, k, λ) designon the set V = {1, 2, . . . , v} with b blocks can be consideredas a decomposition of λKv by b copies of Kk .

I In other words, this is equivalent to considering the existenceof a homomorphism σ ∈ Home(∪bi=1Kk ,Kv) for whichMσ = M

σ= λ. It is easy to see that for such a

homomorphism one has r = Sσ = Sσ

= bkv .I To prove Fisher’s inequality, assume that b < v. Then by Part

(a) for the bth eigenvalue we should have k ≤ λr v. But sincer(k − 1) = λ(v − 1) holds for any 2-design, we should haveλ ≥ r and consequently, v = k.

I Hence, v > k ⇒ b ≥ v.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Petersen and C5!

I Let us consider a homomorphism σ ∈ Hom(P,C5). First, notethat since P contains two disjoint copies of C5 , we candeduce that Sσ = 2.

I Since C5 is critical, the homomorphism should be inHome(P,C5), and it is easy to see that Mσ ≥ 3.

I Applying Part (b) for k = 3 shows that we should have

1.5 ≤ MσSσ

≤ 1.38,

which is a contradiction.

I Hence,

Hom(P,C5) = Home(P,C5) = ∅

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Petersen and C5!

I Let us consider a homomorphism σ ∈ Hom(P,C5). First, notethat since P contains two disjoint copies of C5 , we candeduce that Sσ = 2.

I Since C5 is critical, the homomorphism should be inHome(P,C5), and it is easy to see that Mσ ≥ 3.

I Applying Part (b) for k = 3 shows that we should have

1.5 ≤ MσSσ

≤ 1.38,

which is a contradiction.

I Hence,

Hom(P,C5) = Home(P,C5) = ∅

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Petersen and C5!

I Let us consider a homomorphism σ ∈ Hom(P,C5). First, notethat since P contains two disjoint copies of C5 , we candeduce that Sσ = 2.

I Since C5 is critical, the homomorphism should be inHome(P,C5), and it is easy to see that Mσ ≥ 3.

I Applying Part (b) for k = 3 shows that we should have

1.5 ≤ MσSσ

≤ 1.38,

which is a contradiction.

I Hence,

Hom(P,C5) = Home(P,C5) = ∅

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Petersen and C5!

I Let us consider a homomorphism σ ∈ Hom(P,C5). First, notethat since P contains two disjoint copies of C5 , we candeduce that Sσ = 2.

I Since C5 is critical, the homomorphism should be inHome(P,C5), and it is easy to see that Mσ ≥ 3.

I Applying Part (b) for k = 3 shows that we should have

1.5 ≤ MσSσ

≤ 1.38,

which is a contradiction.

I Hence,

Hom(P,C5) = Home(P,C5) = ∅

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

Petersen and C5!

I Let us consider a homomorphism σ ∈ Hom(P,C5). First, notethat since P contains two disjoint copies of C5 , we candeduce that Sσ = 2.

I Since C5 is critical, the homomorphism should be inHome(P,C5), and it is easy to see that Mσ ≥ 3.

I Applying Part (b) for k = 3 shows that we should have

1.5 ≤ MσSσ

≤ 1.38,

which is a contradiction.

I Hence,

Hom(P,C5) = Home(P,C5) = ∅

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

A spectral bound for the fractionalchromatic number!

I Note that the automorphism group of KG(m,n) actstransitively on both vertices and edges.

I Also, the smallest and largest eigenvalues of the adjacencymatrix of the Kneser graph KG(m,n) are −

(m−n−1

n−1)

and(m−n

n

), respectively.

I Let χf(G) = mn be the fractional chromatic number of the

graph G and also let |V (G)| = ν and d = 2|E(G)|ν .I Applying Part (c) yields,

χf(G) ≥

λG

ν

λGν− d

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

A spectral bound for the fractionalchromatic number!

I Note that the automorphism group of KG(m,n) actstransitively on both vertices and edges.

I Also, the smallest and largest eigenvalues of the adjacencymatrix of the Kneser graph KG(m,n) are −

(m−n−1

n−1)

and(m−n

n

), respectively.

I Let χf(G) = mn be the fractional chromatic number of the

graph G and also let |V (G)| = ν and d = 2|E(G)|ν .I Applying Part (c) yields,

χf(G) ≥

λG

ν

λGν− d

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

A spectral bound for the fractionalchromatic number!

I Note that the automorphism group of KG(m,n) actstransitively on both vertices and edges.

I Also, the smallest and largest eigenvalues of the adjacencymatrix of the Kneser graph KG(m,n) are −

(m−n−1

n−1)

and(m−n

n

), respectively.

I Let χf(G) = mn be the fractional chromatic number of the

graph G and also let |V (G)| = ν and d = 2|E(G)|ν .I Applying Part (c) yields,

χf(G) ≥

λG

ν

λGν− d

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

A spectral bound for the fractionalchromatic number!

I Note that the automorphism group of KG(m,n) actstransitively on both vertices and edges.

I Also, the smallest and largest eigenvalues of the adjacencymatrix of the Kneser graph KG(m,n) are −

(m−n−1

n−1)

and(m−n

n

), respectively.

I Let χf(G) = mn be the fractional chromatic number of the

graph G and also let |V (G)| = ν and d = 2|E(G)|ν .

I Applying Part (c) yields,

χf(G) ≥

λG

ν

λGν− d

.

Random Walks On Graphs: Theory & Applications

Graph homomorphisms and random walks

Some applications

A spectral bound for the fractionalchromatic number!

I Note that the automorphism group of KG(m,n) actstransitively on both vertices and edges.

I Also, the smallest and largest eigenvalues of the adjacencymatrix of the Kneser graph KG(m,n) are −

(m−n−1

n−1)

and(m−n

n

), respectively.

I Let χf(G) = mn be the fractional chromatic number of the

graph G and also let |V (G)| = ν and d = 2|E(G)|ν .I Applying Part (c) yields,

χf(G) ≥

λG

ν

λGν− d

.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some connections

The main setup we applied in this talk has many other variants,e.g.

I (Conservation of energy v.s. Probability) Potential theory andPhysics.

I (Rate of convergence and mixing times v.s. Heat kernel)Analysis and Probability theory.

I (Geometry of homogeneous spaces) Graph theory, Numbertheory, Hyperbolic geometry.Specially, harmonic analysis on reductive groups.

I (Group presentations) Hyperbolic geometry, Combinatorics,3-manifolds.

I (Expanders and rapid mixing) Computer science,Cryptography, Graph theory, Communication theory.

Random Walks On Graphs: Theory & Applications

Epilogue

Some questions!

I (Homomorphism transformations) Develop transformationmethods that makes the inequalities more applicable.

I (Infinite enumerable case) Develop limit theorems such thatone can use inequalities asymptotically for infinite graphs.

I (Group representations) Consider some well known classes ofgroups for which the character table is known and study thecorresponding colouring numbers.

I (Nodal domains) Study nodal domains and the correspondingoptimization problems.

Random Walks On Graphs: Theory & Applications

Epilogue

Some questions!

I (Homomorphism transformations) Develop transformationmethods that makes the inequalities more applicable.

I (Infinite enumerable case) Develop limit theorems such thatone can use inequalities asymptotically for infinite graphs.

I (Group representations) Consider some well known classes ofgroups for which the character table is known and study thecorresponding colouring numbers.

I (Nodal domains) Study nodal domains and the correspondingoptimization problems.

Random Walks On Graphs: Theory & Applications

Epilogue

Some questions!

I (Homomorphism transformations) Develop transformationmethods that makes the inequalities more applicable.

I (Infinite enumerable case) Develop limit theorems such thatone can use inequalities asymptotically for infinite graphs.

I (Group representations) Consider some well known classes ofgroups for which the character table is known and study thecorresponding colouring numbers.

I (Nodal domains) Study nodal domains and the correspondingoptimization problems.

Random Walks On Graphs: Theory & Applications

Epilogue

Some questions!

I (Homomorphism transformations) Develop transformationmethods that makes the inequalities more applicable.

I (Infinite enumerable case) Develop limit theorems such thatone can use inequalities asymptotically for infinite graphs.

I (Group representations) Consider some well known classes ofgroups for which the character table is known and study thecorresponding colouring numbers.

I (Nodal domains) Study nodal domains and the correspondingoptimization problems.

Random Walks On Graphs: Theory & Applications

Epilogue

Some questions!

I (Homomorphism transformations) Develop transformationmethods that makes the inequalities more applicable.

I (Infinite enumerable case) Develop limit theorems such thatone can use inequalities asymptotically for infinite graphs.

I (Group representations) Consider some well known classes ofgroups for which the character table is known and study thecorresponding colouring numbers.

I (Nodal domains) Study nodal domains and the correspondingoptimization problems.

Random Walks On Graphs: Theory & Applications

The End

The End.

Thank You!

IntroductionOutlineThe main setupNatural random walks on graphsRandom walks and LaplacianSummary of part I

Graph homomorphisms and random walksSome important problems in combinatoricsA couple of resultsSome applications

EpilogueThe End