Network Characterisation and Similarity

Richard C. Wilson Dept. of Computer Science

University of York

Outline

1. Brief recap of spectral graph theory

2. Spectral Similarity

3. Random Walks and Differential Equations

4. Quantum Graphs

Notation

Common notation

V is the set of vertices (|V| is the order of the graph)

E is the set of edges (|E| is the size of the graph)

X is an attribute functions, maps vertices and edges onto

their attributes

),,( XEVG

edge directed ,,,

edge undirected ,,,

VvVuvue

VvVuvueEe

Graph Spectrum

Matrix Representation

A Matrix Representation X o f a n e t w o r k i s m a t r i x w i t h e n t r i e s

r e p r e s e n t i n g t h e v e r t i c e s a n d e d g e s

A d j a c e n c y

00110

00100

11010

10101

00010

5

4

3

2

1

5 4 3 2 1

A1

2

3

4

5

20000

01000

00300

00030

00001

DDegree matrix

Matrix Representation

The Laplacian (L) is

Signless Laplacian

20110

01100

11310

10131

00011

ADL

ADL s

Matrix Representation

Normalized Laplacian

Entries are

2

1

2

1

2

1

2

1

ˆ

LDD

ADDIL

otherwise0

),(1

1

ˆ Evudd

vu

Lvu

uv

Incidence matrix

The incidence matrix of a graph is a matrix describing the relationship

between vertices and edges

Relationship to signless Laplacian

Adjacency

Laplacian

10

11

01

2,32,1

M1

2

3

DMMA T

TMMDL 2

T

s MML

Matrix Representation

Consider the Laplacian (L) of this network

Clearly if we label the network differently, we get a different matrix

In fact

represents the same graph for any permutation matrix P of the n labels

20110

01100

11310

10131

00011

5

4

3

2

1

5 4 3 2 1

1

2

3

4

5

TPLPL '

1

2

20101

01100

11301

00011

10113

5

4

3

2

1

5 4 3 2 1

Characterisations

Are two networks the same? (Graph Isomorphism), or is

there a bijection between the vertices such that all the edges

are in correspondence?

Interesting problem in computational theory, complexity

unknown but hypothesised as separate class in NP-

hierarchy, GI-hard

Graph Automorphism: Isomorphism between a graph and

itself. There is an equivalence between GI and counting

number of GAs

G1 G2

G1 G2 G1

G2

2121 ##2# GIGIGGI

Characterisations

An equivalent statement: Two networks are isomorphic iff

there exists a permutation matrix P such that

X should contain all information about the network

– Applies to L, A etc not to D

P is a relabelling; changes the order in which we label the

vertices

Our measurements from a matrix representation should be

invariant under this transformation (similarity transform)

TPPXX 12

X is a full matrix

representation

Spectral Graph Theory

Properties of the graph from the eigenvalues (eigenvectors) of

a matrix representation of the graph

1

UUUUX

UUX

T

LR

TSymmetric (undirected)

Non-symmetric

Perron-Frobenius Theorem

Perron-Frobenius Theorem:

If X is an irreducible square matrix with non-negative entries, then there exists an eigenpair (λ,u) such that

Applies to both left and right eigenvector

•Key theorem: if our matrix is non-negative, we can find a principal(largest) eigenvalue which is positive and has a non-negative eigenvector

•Irreducible implies associated digraph is strongly connected

0

j

i

u

i

R

Spectrum

The graph has a ordered set of eigenvalues (λ0, λ1,… λn-1)

in terms of size (I will use smallest first).

The (ordered) set of eigenvalues is called the spectrum of

the graph.

Theorem: The spectrum is unchanged by the relabelling

transform

Corollary: If two graphs are isomorphic, they have the

same spectrum

This does not solve the isomorphism problem, as two

different graphs may have the same spectrum

12

12

TPPXX

Spectrum

These two graphs have the same spectrum using the

Laplacian representation

This is a cospectral pair. Necessary but not sufficient to

determine isomorphism.

The matrix representation we use has a big effect on how

many of these cospectral graphs there are

5.24

[3]2

2

0.76

5.24

[3]2

2

0.76

Cospectral graphs

How many such graphs are there and how does it depend

on representation? (Zhu & Wilson 2008)

*50 trillion

graphs of size

13

*

Cospectrality

Open problem: Is there a representation in which nearly all

graphs are determined by the spectrum (non-cospectral)?

Answer for trees: No, nearly all trees are cospectral

In practice, cospectrality not a problem since two randomly

selected graphs have tiny chance of being cospectral.

If we pick graphs from a specialised family, may be a

problem, for example regular, strongly regular graphs

Spectrum of A

Spectrum of A: Positive and negative eigenvalues

Bipartite graph: If λ is an eigenvalue, then so is –λ, Sp(A)

symmetric around 0

Perron-Frobenius Theorem (A non-negative matrix)

n-1 is largest magnitude eigenvalue, corresponding

eigenvector xn-1 is non-negative

01

max110max 0

0

n

n

i

dd

Spectrum of L

Spectrum of L: L positive semi-definite

There always exists an eigenvector 1 with eigenvalue 0,

because of zero row-sums

The number zeros in the spectrum is the number of

connected components of the graph.

n

E

n

i

1100

2

Spectrum of L

A spanning tree of a graph is a tree containing only edges

in the graph and all the vertices

Example

Kirchhoff’s theorem

The number of spanning trees of a graph is

1

1

1 n

i

in

Spectrum of normalised L

Spectrum of : Positive semi-definite

As with Laplacian, the number zeros in the spectrum is the

number of disconnected components of the graph.

Eigenvector exists with eigenvalue 0 and entries

‘scale invariance’

20 110

n

i V

L

Tnddd 21

References

Spectra of Graphs, Brouwer & Haemers, Springer

Graph Spectra for Complex Networks, Van Mieghem,

Cambridge University Press

Spectral Graph Theory, Fan Chung, American

Mathematical Society

Coding Attributes

So far, we have considered edges only as present or absent

{0,1}. If we have more edge information, can encode in a

variety of ways. Edges can be weighted to encode

attributes, include diagonal entries to encode vertices

00110

00100

11010

1016.02.0

0002.04.0

A0.4

0.6

0.2

Coding Attributes

•Note: When using Laplacian, add diagonal elements after

forming L

•Label attributes: Code labels into [0,1]

•Example: chemical structures

Edges

─ 0.5

═ 1.0

Aromatic 0.75

Vertices

C 0.7

N 0.8

O 0.9

Coding Attributes

Spectral theory works equally well for complex matrices

Matrix entry is x+iy so can encode two independent

attributes per entry, x and y. Symmetric matrix becomes

Hermitian matrix

Eigenvalues real, eigenvectors complex

03.01.00

2.01.003.05.0

03.05.00

†

i

ii

i

A

AA

Coding Attributes

Example: Shape skeletons

Shock graph has vertices where shocks meets and edges

with lengths l and angles θ

Encode as complex weight

Naturally hermitian as

ijijijij

i

ijij illelA ij

sincos

jiij

jiij ll

Similarity

Similarity of Networks

How can we measure the similarity of two networks?

Key idea: Graph Edit Distance(GED) uses edit operations

– Vertex insertion, deletion

– Edge insertion, deletion

– Relabelling a vertex

Associate a cost with each operation. Find a sequence of

edit operations which transforms one network into the

other. The minimum possible cost of a sequence is the

graph edit distance.

NP-complete so we cannot actually compute it

GED - example

Edge deletion

Cost ed Vertex deletion

Cost vd

Edge insertion

Cost ei

Vertex relabel

Cost vl

G1

G2

The sequence of edit

operations is an edit path

E

c(E)=ed+vd+ei+vl

)(min),( 21 EcGGGEDE

Graph similarity

The simplest form of GED is zero cost for vertex

operations and relabelling

Then equivalent to Maximum Common Subgraph [Bunke,

PAMI 1999]

Since we cannot compute GED, we generally resort to

approximate methods, either compute matches or compare

features. If we can get good features, we can use them to

compare graphs

Eigenperturbation

Let Y=X+N where X and Y are two similar networks and

N is small

Since the change in eigenvalues is

bounded by the difference between X and Y

NXYN 1 nkkk

Fn NN 1

kj

j

jk

k

T

j

kk

k

T

kkk

uNuu

uu

Nuu

Spectral Similarity

How good is the spectrum for similarity comparisons?

[Zhu, Wilson 2008]

Theorem: The eigenvector components are permuted by the

relabelling transform

The columns of U are ordered by the eigenvalues, but the

rows still depend on the labelling

Eigenvectors

12

1122

12

PUU

PUPUUU

PPXX

TTT

T

Non-uniqueness of U

The eigenvectors of a graph are not unique. They have a

sign ambiguity, if u is an eigenvector, so is –u

Need to apply sign correction

Repeated eigenvalues makes the situation worse: If (,u1)

and (,u2) are eigenpairs, then so is (,au1+bu2)

)()( uuXuXu

212121 uuXuXuuuX bababa

Spectral Features

Theorem:

All graphs which have simple spectra can be distinguished

from each other in polynomial time

Simple spectrum means than there are no repeated eigenvalues in the

spectrum, hence the eigendecomposition is unique (up to signs).

Then we can order the components of the eigenvectors in polynomial

time (for example by sorting)

Open Problem: Repeated eigenvalues, difficult

graphs for isomorphism and labelling ambiguity

are all connected in a way not yet understood

Random walks, Diffusions and Differential Equations

Random Walks

•Spectral features are not tightly coupled to structure

•Can we explore the structure of the network?

•A random walker travels between vertices by choosing an

edge at random

•At each time step, a step is taken down an edge

Discrete Time Random Walk

Imagine that we are standing at vertex ui

At each time, we chose one of the available edges with equal

probability

Then the probability of arriving at vertex uj is

Therefore, at the next time step, the distribution is

i

ij

ji

ji

iijd

A

Euu

EuuduuP

),( 0

),( 1

)|(

i

it

i

ij

i

itijjt

uPd

A

uPuuPuP

)(

)()|()(1

Discrete Time Random Walk

T is the transition matrix of the walk, a stochastic matrix

(rows sum to 1). Perron-Frobenius theorem implies largest

magnitude eigenvalue 1.

If we start in state π0 then at time t

)(

)()(

1

1

1

1

1

ADTTππ

ADππ

tt

tt

i

it

i

ij

jt uPd

AuP

t

t Tππ 0

Discrete Time Random Walk

What happens after a very long time?

t

ts Tππ 0lim

T

L

t

n

t

t

R

T

L

t

R

tT

UU

UU

1

1

0

0

0

0lim

1||

t

t

1lim

1

t

t

T

LR UUT

T

nL

nRn

T

nLnRns

1,

1,11,1,01

11

,1

u

1uuuππ

Discrete Time Random Walks

After a very long time, the walk becomes stationary Only the

largest (left) eigenvector of T survives

This is the principal eigenvector of T (with λ=1) and is easy

to solve; it is

After a long time, we are at each node with a probability

proportional to its degree. It is natural to think of the

probability as a measure of centrality. In this situation,

eigenvector centrality (of T) coincides with degree centrality

πTπ

||2 E

dπ i

i

PageRank

One important application for random walks is for the web

More central pages are more important

Idea:

Surfer clicks links to new pages at random

May also quit and start fresh at a random page (‘teleporting’)

Importance of page is prob of ending up there

Links are directed, but makes no difference to the formulation

J is matrix of all-ones (teleportation transitions)

α is the probability of starting over

Eigenvector centrality for T is the PageRank (Google) of each

page

JADT 1)1(

Backtrackless Walks

The random walk suffers from the problem of tottering

This reduces expressive power and masks structural differences.

An alternative is to use a backtrackless walk where reverse steps

are not allowed.

We can treat backtrackless walks in much the same way by using a

graph transformation.

Backtrackless walks

•(Oriented) Line graph:

1 2

3 4

e21

e12

e23 e32 e42

e24

e41 e14

e43

e34

e21

e12

e23 e32

e42

e24

e41

e14

e43

e34

e23

e21

e12

e32

e42

e24

e41 e14

e43

e34

Line graph (LG): Random walk Oriented Line graph (OLG):

Random walk, no backtracking

Backtrackless walk

The adjacency matrix of the OLG is

A random walk using this transition matrix is a

backtrackless walk on the original graph.

The size of Q is ~|E|2 so potentially very large but sparse.

The mathematical structure allows us to find efficient ways

of evaluating properties of the walk.

[Backtrackless Walks on a Graph, Aziz, Wilson, Hancock 2012, IEEE

TNNLS]

otherwise 0

,,),(),,( if 1)],(),,[(

dacbEdcbadcbaQ

Differential Equations

Differential Equations on Graphs

A whole host of important physical processes can be

described by differential equations

Diffusion, or heat flow

Wave propagation

Schrödinger Equation

pt

p 2

pt

p 2

2

2

Vppmt

pi

22

2

Laplacian

is the Laplacian differential operator

In Euclidean space

Different in non-flat spaces

Take a 1D discrete version of this

i, i-1,i+1 denote neighbouring points

2

2

2

2

2

22

zyx

2

)(2)()(

)()()()(

)()(

11

11

2/12/12

22

iii

iiii

ii

xpxpxp

xpxpxpxp

xx

px

x

p

x

pp

xi xi+1 xi-1

Laplacian

A graph which encodes the neighbourhood structure

The Lapacian of this graph is

Apply L to a vector (a ‘function’ taking values on the vertices)

So the graph Laplacian is a discrete representation of the calculus Laplacian

– Vertices are points in space

– Edges represent neighbourhood structure of space

– Note minus sign!

p

ppp iiii

2

11 2

Lp

i i+1 i-1

110

121

011

L

Diffusion

On a network, we identify the Laplacian operator 2 with

the Laplacian of the network L

Discrete space, continuous time diffusion process

Lpp

t

-L 2

pt

p 2

Heat Kernel

Solution

Heat kernel H(t)

Hij(t) describes the amount of heat flow from vertex i to j at

time t

Essentially another matrix representation, but can vary time

to get different representations

)exp()(

)0()()(

tt

tt

LH

pHp

)0()()0()exp(

)0()exp(

)0()()0()(

pLHpLL

pL

pHpH

tt

tt

tt

tt

)()(

tt

tLp

p

Diffusion as continuous time random walk

Consider the following walk on a k-regular graph

At each time step:

stay at the same vertex with probability (1-s)

Move with prob. s to an adjacent vertex chosen uniformly at random

This is called a lazy random walk

Transition matrix

LI

AII

AIT

k

s

kk

s

kss

)(

1)1(

Diffusion as continuous time random walk

Let s be a time-step

n=t/s is the number of steps to reach time t

n

n

n

nk

t

k

s

t

LI

LI

TT )(

LLIT

k

t

nk

tt

n

nsexplim)(lim

0

Small times

Large times

Only smallest eigenvalues survive, λ1=0 and λ2

Behaves like Fiedler vector (Spectral cut)

Spectral representation

t

t

T

e

e

t

ttt

2

1

0

0

)exp(

)exp()exp()(

UULH

t

ttt

LI

LLIH

22

!2

1)(

0 te

Tte

nt 22

21

)( JH

In all but some special cases, the heat kernel can be

reconstructed from what happens at the vertices only (the

self-heat) H(u,u)

Sun et al used diagonal elements of the heat kernel to

characterise 3D object meshes

[A Concise and Provably Informative Multi-Scale Signature Based on Heat

Diffusion, Sun, Ovsjanikov ,Guibas, Comput. Graph. Forum, 2009]

Describes a particular vertex (for matching) by heat content at

various times. Times carefully chosen to be informative

about the diffusion process.

Heat Kernel Signature

]),,(),,(),,([HKS210

xxHxxHxxH ttt

A global version describing the whole network can be

constructed by histogramming over the vertices

),,(),,(hist

),,(),,(hist

GHKS 2211

2211

11

00

uuHuuH

uuHuuH

tt

tt

Example: Financial network

Analysis of stock closing prices over a long period of time.

Correlation analysis reveals connections between stocks.

Threshold to get a complex network.

Analysis of the structure of the network through graph

signatures reveals interesting features.

[Matlab plot]

Centrality

We can use the heat kernel to define another node centrality

measure. Consider the following adjacency matrix as a

weighted graph (with weights 1/√dudv on the edges)

The weighted sum of all paths of length k between two

vertices u and v is given by

2

1

2

1

ˆ

ADDA

uv

kA

Subgraph Centrality

Total communication between vertices is sum over paths of

all lengths

α allows us to control the weight of longer paths vs shorter

What should α be?

Number of possible ways to go increases factorially with k

Longer paths should be weighted less

0

ˆ

k

uv

k

uvC A

0

ˆ!k

uv

kk

uvk

tC A

Subgraph centrality

Subgraph centrality (Estrada, Rodríguez-Velázquez 2005):

centrality is the ability of vertex to communicate with others

Relationship to heat kernel

Subgraph centrality generally uses A, but results coincide

exactly for regular graphs

v k

uv

kk

v

uvuk

tCs

0

ˆ!

A

H1s

CA

H

A

AIL

V

e

ek

te

ee

eet

t

k

tkk

t

tt

ttt

0

ˆ

ˆˆ

ˆ!

)(

Wave propagation

Wave equation in discrete space, continuous time

The wave kernel (analogous to the heat kernel) is

pt

p 2

2

2

Lpp

2

2

t

LW itt exp)(

Features of wave kernel

No dissipation: waves continue travelling and no steady

state at large times

Basic waves transmitted are the eigenvectors of L and the

frequencies are √

What happens at small times t=δ?

All vertices reached with small amplitude at any non-zero

time (infinite propagation speed)

Tititt UULW expexp)(

22

2

1)( LLW iI

Wave Kernel Signature

We can define a wave kernel signature analogous to the

heat kernel signature:

A global signature of the graph can be constructed again

with histograms.

[The wave kernel signature: A quantum mechanical approach to shape

analysis,Aubry, Schlickewei, Cremers, ICCV Workshops, 2011]

k

kk

euveu

2

2

2

)log(exp)(),(WKS

Example: PPIs

PPIs often modelled with geometric model of complex

networks

Biogrid human PPI

|V|=1923, |E|=3866, density 0.00209, <k>=4.02

What model fits this network? Use Monte-carlo sampling

of model and examine similarities between samples and

between sample and PPI

Example: PPIs

PPI-model model-model

Wave kernel signature similarity [Graph Signatures for Evaluating Network Models, Wilson, ICPR 2014]

GEO3D Configurational

Quantum Graphs

Quantum Graph

Free space Schrodinger equation pmt

pi 2

2

2

Infinite boundary conditions

Constrained

More constrained

Quantum graph

1D piece of space

Graph Calculus

Key Idea: Geometric graph

Now the graph lives in two spaces

Node-space V: Point-like measure

Edge-space E: interval (Lebeguese measure)

Graph functions exist on vertices and on edges

Called a quantum graph or metric graph

Edge: interval

or length

Node: Point-like

Graph Laplacian

To solve problems like wave propagation on these graphs,

we need to know the Laplacian. The edge part is like a

normal 1D Laplacian along the edges

The vertex part is essentially the same as a type of

discrete Laplacian, except that it is adjacent to the

edges where they meet, not other vertices

eue

eueV ufu

f,

, )()(

1n

V

2

2

dx

fdE

Vertex- and Edge-functions

A function f on the graph exists both at vertices and along

the edge intervals.

If Ef=0 then the function is said to be vertex-based

Similarly Vf=0 gives an edge-based function

Ef=0 implies the function is linear on edges

In this case we have

– f exists on vertices

– f is linear between vertices

– Continuity means f is fully defined by the vertex values

– The Laplacian is a discrete Laplacian

•The new and old frameworks co-incide when edge

functions are linear

Edge-based functions

Edge-based functions are more interesting and complex.

Depend on boundary conditions imposed at the vertices.

Most natural and interesting case are Neumann conditions

Sum of inward pointing gradients must be zero at vertices. f

must also be continuous. Given these conditions, what are

the eigenvalues and eigenfunctions of the graph

Laplacian?

0)()1(

0

,

,

1 ,

eue

ue

x

V

xf

f

ue

Eigenvalues

Assumption: All edge lengths are constant and equal. f

must satisfy

B and C to be determined from the Neumann boundary

conditions

)(cos)(),( eBxeCxef ee

fdx

fd

2

2

Edge-based Eigenfunctions

A is the adjacency matrix of the (combinatorial) graph

B(e) and C(e) are determined by the corresponding eigenvector

In particular, these eigenfunctions take the value of the

eigenvector on the vertices (vertex-supported)

Behave like eigenvectors of (related to random walk)

EΔ

,-λ

of eigenvaluean is

cos

then,11 and of eigenvaluean is If

matrix)n (Transitiomatrix adjacency normalised Row

1

1

T

ADT

T

Edge-interior Eigenfunctions

The remaining eigenfunctions are edge-interior (zero on all

vertices) and have eigenvalues or 2

There are two types

Eigenvalue has condition that the gradients sum to zero at

each vertex (symmetric)

•Eigenvalue 2 has condition that the directed gradients

sum to zero, accounting for the direction of the edge

(antisymmetric)

•B(e) is /2, C(e) must be determined from the conditions

2cos)(),(

ee xeCxef

Edge-interior Eigenfunctions

Theorem: Let Q be the adjacency matrix of the OLG. Then

the eigenvectors s with eigenvalue -1 of Q have

components equal to the values of C(e) for the symmetric

eigenfunctions. The eigenvectors with eigenvalue +1 have

components equal to C(e) in the antisymmetric case.

The structure of these eigenfunctions are determined by

certain eigenvectors of Q (related to the backtrackless

walk)

Eigenfunction summary

Two sorts of eigenfunction of the edge-based Laplacian

Vertex supported

– Determined by the eigensystem of the row-normalized adjacency

matrix, or equivalently the structure of the random walk

Edge interior

– Determined by selected eigenvectors of the OLG, i.e. the structure

of the backtrackless random walk

The edge-based Laplacian has a rich structure from both

types of walk

Applications

Wave equation:

With discrete Laplacian L, signal recieved

instantly (no ‘length’ between vertices) but the

edge-based Laplacian has finite speed of

transmission

ft

f

2

2

Wave Equation

A better model of transmission in networks?

Evolution of Gaussian wave packet on a graph

Evolution of Gaussian wave packet on a graph with 5 vertices and 7 edges

Wave packet signature

Local signature for edge sampled at different

times

Global signature histograms over edges

ntutututuWPS ,,...,,,,,,)( 210

||321 ,...,,,hist)( EWPSWPSWPSWPSGGWPS

Unweighted Graphs

Truncated

Laplacian Wave packet

Signature

Method

Delaunay

Triangulatio

n

Gabriel

Graphs

RN

Graphs

Wave Packet Signature 0.9965 0.9511 0.8235

Random Walk Kernel 0.9526 0.9115 0.8197

Ihara Coefficients 0.9864 0.8574 0.7541

References

Eigenfunctions of the edge-based Laplacian on a graph, Wilson, Aziz,

Hancock,, Linear Algebra and its Applications, 2013

Edge-based Operators for Graph Characterization. Aziz,

Furqan (2014) PhD thesis, University of York.

http://etheses.whiterose.ac.uk/6218/

Graph characterization using Gaussian wave packet signature, Aziz,

Wilson, Hancock, LNCS, 2013

Wave equations on graphs and the edge-based Laplacian, Friedman, .

Tillich, Pacific Journal of Mathematics, 2004

Extra slides

General Solution of the Wave Equation

•With edge coordinate χ and time t, edge-based wave equation is

•Seek separable solutions of the form

.

with edge-based eigenfunctions

•Gives a temporal solution

•By superposition, we obtain the general solution

tu

t

tuE ,

,2

2

tntn

xnxeBeCtu

nn

n

2sin2cos

2,cos,,

,,

tgtu n ,,

xnxeBeCn 2,cos,,

tntntg nn 2sin2cos ,,

Initial Conditions

•Since the wave equation is second order partial differential equation, we can impose initial conditions on both position and speed

and we obtain

We can find these coefficients using the orthonormality of eigenfunctions. So we get

where

similarly

where

1

0

2

, ),( nixiiB

n eexedxqeG

e

nnn GGeCnx *

,,,2

1,2

1

0

2

, ),( nixiiB

n eexedxpeF

e

nnn FFeC *

,,,2

1,

n

n xnxeBeCnxq 2,cos,2,

n

n xnxeBeCp 2,cos,,

qt

u

0,

pu 0,

Gaussian Wave Packet

•We assume the initial position be a Gaussian wave packet

p(e,x)=exp{-a(x-μ)2}

• on one particular edge and zero everywhere else. Then we have

•Solving, we get

•And so

•Similarly

nxiaixa

aiiB

n edxeeeeF

2

2

24,

2

224

1

2

,

nanBi

n eea

F

nBeCe

a

na

n

2cos,2

24

1

,

nBeCe

a

na

n

2sin,2

24

1

,

General Solution of the Wave Equation

where Ωa represents the set of vertex-supported eigenvalues and Ωb and Ωc represent the set of edge-interior eigenvalues. i.e., π and 2π. Also

Solution of wave equation with Gaussian wave packet as initial condition

Cospectral Graphs

Wavepacket Signature