Basic graph theory
18.S995 - L19
dunkel@math.mit.edu
no cycles
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Isomorphic graphs
f(a) = 1 !
f(b) = 6!
f(c) = 8!
f(d) = 3!
f(g) = 5!
f(h) = 2!
f(i) = 4!
f(j) = 7
image source: wiki
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Complete simple graphs on n vertices
K1
K2
K3
K4
K5
K6
K7
K8
K9
K10
K11
K12
Bi-partite graph
Planar, non-planar & dual graphs
(a)
(b)
(c)
(d)
Figure 1.2: Planar, non-planar and dual graphs. (a) Plane ‘butterfly’graph. (b, c) Nonplanar graphs. (d) The two red graphs are both dual to the blue graph but they are not
isomorphic. Image source: wiki.
Given a graph G, its line graph or derivative L[G] is a graph such that (i) each vertex
of L[G] represents an edge of G and (ii) two vertices of L[G] are adjacent if and only if
their corresponding edges share a common endpoint (‘are incident’) in G (Fig. ??). This
construction can be iterated to obtain higher-order line (or derivative) graphs.
1.3
Adjacency and incidence
Algebraic characterization
|V|x|V| matrix
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connected by an edge. The adjacency matrix A(G) = (Aij ) is a |V | ⇥ |V |-matrix that lists
D(G) = diag deg(v1 ), . . . , deg(v|V | )
(1.4)
all the connections in a graph. If the graph is simple,
then A is symmetric and has only
Characteristic
polynomial
0
1
For the graph in Fig. 1.3a, one has
3 0 0 0 0
C we have
entries 0 or 1. For example, for theB
0 2 in
0 Fig.
0 01.3a,
Bgraph
C
0
1
B
D = B0 00 12 10 10C
(1.5)
C0
@0B 0 0 3 0AC
B10 00 00 02 1C
0
C
A=B
(1.1)
B1 0 0 1 0 C
@1 characteristics
0 1 0 1A of random graphs, and we will
The
degree
distribution
is
an
important
(a)
(b)
(c)
(d)
0
1
0
1
0
return to this topic further below.
~
If
is simple,
directed,then
we the
maydiagonal
still define
a signed
adjacency
If the
the graph
graph is
elements
of A
are zero.matrix A with elements
8
The columnof(row)
sum
definesThese
the degree
(connectivity)
of the vertex
Figure 1.3: Construction
a line
graph.
figures
show
(a, with blue vertices)
>
from
vi toavjgraph
< 1, if edge goes
X
and its line graph (d, with green
vertices).
Each
vertex
thevi line graph is shown(1.6)
labeled
~
Aij = +1, deg
if edge
vj to
(vi ) goes
= from
Aij of
(1.2)
>
:
with the pair of endpoints of the corresponding
in the original graph. For instance,
j
0,
otherwiseedge
the green vertex
on volume
the right
labeled
corresponds
to the edge on the left between the
and the
of the
graph is1,3
given
by
The characteristic polynomial of a graph is defined as the characteristic polynomial of
X
X other green vertices: 1,4 and 1,2
blue vertices
1
and
3.
Green
vertex
1,3
is
adjacent
to
three
the adjacency matrix
vol(G) =
deg (vi ) =
Aij
(1.3)
(corresponding to edges sharing the endpointV 1 in the blue
graph) and 4,3 (corresponding
ij
= det(A xI)
(1.7)
to an edge sharing the endpoint 3 in p(G;
the x)
blue
graph). Image and text source: wiki.
The degree matrix D(G) is defined as the diagonal matrix
For the graph in Fig. 1.3a, we find
D(G) = diag deg(v1 ), . . .2, deg(v
(1.4)
4 |V | )
p(G; x) = x(4 2x 6x + x )
(1.8)
For the graph in Fig. 1.3a, one has
Characteristic polynomials are not diagnostic
for graph
0
1 isomorphism, i.e., two noniso3 0 0 0polynomial.
0
morphic graphs may share the same characteristic
B0 2 0 0 0C
B
C
B
D = B40 50 2 0 0C
(1.5)
C
@0 0 0 3 0A
0 0 0 0 2
The degree distribution is an important characteristics of random graphs, and we will
return to this topic further below.
~ with elements
If the graph is directed, we may still define a signed adjacency matrix A
Adjacency matrix
Nauru graph
“integer graph”
|V|x|V| matrix
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Complexity
!
Basic operations in a graph are:
1.
Adding an edge
2.
Deleting an edge
3.
Answering the question “is there an edge between i and j”
4.
Finding the successors of a given vertex
5.
Finding (if exists) a path between two vertices
Complexity
In case that we’re using adjacency matrix we have:
1.
Adding an edge – O(1)
2.
Deleting an edge – O(1)
3.
Answering the question “is there an edge between i and j” – O(1)
4. Finding the successors of a given vertex – O(n)
5.
Finding (if exists) a path between two vertices – O(n^2)
Complexity
List
While for an adjacency list we can have:
1.
Adding an edge – O(log(n))
2.
Deleting an edge – O(log(n))
3. Answering the question “is there an edge between i and j” – O(log(n))
4.
Finding the successors of a given vertex – O(k), where “k” is the
length of the lists containing the successors of i
5.
Finding (if exists) a path between two vertices – O(n+m) with m <= n
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connected by an edge. The adjacency matrix A(G) = (Aij ) is a |V | ⇥ |V |-matrix that lists
all the connections in a graph. If the graph is simple, then A is symmetric and has only
Degree matrix
(a)
entries 0 or 1. For example, for the graph in Fig. 1.3a, we have
0
1
0 1 1 1 0
B1 0 0 0 1 C
B
C
B
A = B1 0 0 1 0 C
C
@1 0 1 0 1 A
(b)
(c)
0 1 0 1 0
(1.1)
(d)
If the graph is simple, then the diagonal elements of A are zero.
The columnof(row)
sum
definesThese
the degree
(connectivity)
of the vertex
Figure 1.3: Construction
a line
graph.
figures
show a graph
(a, with blue vertices)
X
and its line graph (d, with green vertices).
Each
vertex
labeled
deg (vi ) =
Aij of the line graph is shown (1.2)
with the pair of endpoints of the corresponding edge
in the original graph. For instance,
j
the green vertex
on volume
the right
labeled
corresponds
to the edge on the left between the
and the
of the
graph is1,3
given
by
X
X other green vertices: 1,4 and 1,2
blue vertices 1 and 3. Green vertex 1,3 is adjacent
to three
vol(G) =
deg (vi ) =
Aij
(1.3)
(corresponding to edges sharing the endpointV 1 in the
blue
graph)
and
4,3
(corresponding
ij
to an edge sharing the endpoint 3 in the blue graph). Image and text source: wiki.
The degree matrix D(G) is defined as the diagonal matrix
D(G) = diag deg(v1 ), . . . , deg(v|V | )
For the graph in Fig. 1.3a, one has
0
3
B0
B
D=B
B40
@0
0
0
2
0
0
0
0
0
2
0
0
0
0
0
3
0
1
0
0C
C
0C
C
0A
2
(1.4)
(1.5)
The degree distribution is an important characteristics of random graphs, and we will
return to this topic further below.
~ with elements
If the graph is directed, we may still define a signed adjacency matrix A
0 0 1 1 1 0
Figure 1.2: Planar, non-planar and dual graphs. (a) Plane ‘butterfly’graph. (b, c) Nonplanaryielding
graphs.
The two polynomial
red graphs are both dual to the blue graph but they are not
the(d)
characteristic
isomorphic. Image source: wiki.
2
2
p(L[G]; x) = (x + 2) x + x
1 [(x
3)x
x + 2]
(1.12)
Directed
incidence
In addition
to the undirected
matrix
C, we
Given
a graph
G, its matrix
line graph
or derivative
L[G] is aincidence
graph such
that
(i)still
each vertex
~ as follows
define
a directedan
|V |edge
⇥ |E|-matrix
C
of L[G]
represents
of G and
(ii) two vertices of L[G] are adjacent if and only if
8 a common endpoint (‘are incident’) in G (Fig. ??). This
their corresponding edges share
>
< 1, if edge es departs from vi
construction can be iterated
~ =to obtain higher-order line (or derivative) graphs.
C
(1.13)
is
1.3
>
:
+1, if edge es arrives at vi
0,
otherwise
Adjacency and incidence
For undirected graphs, the assignment of the edge direction is arbitrary – we merely have
~ sum to 0. For the graph in Fig. 1.3a, one
to ensure that the columns s = 1, . . . , |E| of C
Adjacency
matrix Two vertices v1 and v2 of a graph are called adjacent, if they are
finds
0
1ij ) is a |V | ⇥ |V |-matrix that lists
connected by an edge. The adjacency
matrix
A(G)
=
(A
1
1
1 0
0
0
all the connections in a graph.BIf1 the0 graph
A is symmetric and has only
C
0 is1 simple,
0
0 then
B
C
~ =B 0
1
0
0
1 0C
C
(1.14)
B
C
@0
0
1
0
1
1A
0
0
0
1
0
1
6
(a)
(b)
(c)
(d)
Figure 1.3: Construction of a line graph. These figures show a graph (a, with blue vertices)
and its line graph (d, with green vertices). Each vertex of the line graph is shown labeled
with the pair of endpoints of the corresponding edge in the original graph. For instance,
(a)
(b)
A(L[G]) =
C(G)> · C(G)
2I
(c)
,
A(L[G])rs = Cir C(d)
2
is
(1.10)
rs
example in Fig. 1.3, we thus find
1.3.1For the
Laplacian
0 graphs. (a) 1
Figure 1.2: Planar, non-planar and dual
Plane ‘butterfly’graph. (b, c) Non0
1
1
1
0
0
The |V | ⇥ |V |-Laplacian matrix L(G) of a graph G, often also referred to as Kirchho↵
B
planarmatrix,
graphs.
(d) The two red graphs 1.3.1
are both
to the
blue
graph matrix
but they are not
1 Laplacian
0 dual
1 matrix
0C
B1 0 degree
C
is defined as the di↵erence between
and
adjacency
C
isomorphic. Image source: wiki.A(L[G]) = B
B1 1 0 0 1 1 C
(1.11)
B=
matrix L(G) of a graph(1.15a)
G, often also re
0|V |0⇥A
0|V 0|-Laplacian
1C
LThe
B1 D
C
@0 1 1is defined
0 0 1A
matrix,
as the di↵erence between degree matrix and adja
Hencea graph G, its line graph or derivative
0 0 1 1L[G]
1 0 is a graph such that (i) each vertex
Given
L=D A
8
of L[G] represents
edge>
G iand
deg(v
), if(ii)
i =two
j vertices of L[G] are adjacent if and only if
yielding thean
characteristic
polynomial
<of
Hence
their corresponding edges
a
common
(‘are
incident’)
This
8
Lij = share
(1.15b)
1,
if vi 2andendpoint
vj are connected
by edge in G (Fig. ??).
2
>
p(L[G];
x) = (x + 2) x + x 1 [(x 3)x
x+
2] i ), if i = j (1.12)
>
deg(v
: to
<
construction can be iterated
obtain
higher-order
line
(or
derivative)
graphs.
0,
otherwise
Lij =
1,
if vi and vj are connected by edg
>
matrix In addition to the undirected incidence
matrix C, we still
:
As weDirected
shall seeincidence
below, this
matrix provides
an
important
characterization
of the underlying
0,
otherwise
~ as follows
define a directed |V | ⇥ |E|-matrix C
1.3 graph.
Adjacency and8incidence
As we shall see below, this matrix provides an important characteriza
The |V | ⇥ |V |-Laplacian matrix
alsoebe
expressed in terms of the directed incidence
>
ifgraph.
edge
s departs from vi
< 1, can
~matrix
matrix C,
as
+1,v1 ifand
edge
arrives
at
vi
Adjacency
Two C~vertices
ves2|V
of| ⇥a|Vgraph
are called
adjacent,
if they are
is =
The
|-Laplacian
matrix can
also(1.13)
be expressed
in terms of
>
:
~ as = (Aij ) is a |V | ⇥ |V |-matrix that lists
0, >matrix
otherwise
matrix A(G)
C,
connected by an edge. The L
adjacency
~
~
~ ir C
~ jr
=C ·C
,
Lij = C
(1.16)
>
all the connections
in graphs,
a graph.
If the graph
simple,
then
A is~ symmetric
and has only
~we
~ C
~
For undirected
the assignment
of theis
edge
direction
is arbitrary
merely
L=C
·–C
,have L = C
(a)
For
graph
1.3a,s =
one
~ sum to 0. For the graph in Fig. 1.3a, one
to the
ensure
that in
theFig.
columns
1, .finds
. . , |E| of C
0
1 1.3a, one finds
For the graph in Fig.
finds
3
1
1
1 0
0
0B
1C
3
1
1
1
2
0
0
1
1
1
1
0
0
0
B
C
B 1 2
0
B
C
B
B
L =BB1 1 0 0 0 2 1 01 00CC
~ = B@
CC
2
L=B
0
1
0
0
1
0
C
A
B 1 0 (1.14)
1 0
1 3
1C
B
@ 1 0
@0
1
0
1
0
1
1A
0
1 0
1 2
(b)
0
0
0
1
(c)
0
1
0
(d)
1
0
Properties We denote the eigenvalues
of L by We denote the eigenvalues of L by
Properties
6
ij
ir
1
1 0
0 (1.17)
1C
C
1 0C
C
3
1A
1 2
jr
Normalized Laplacian The associated normalized Laplacian L(G) is defined as
L=D
1/2
·L·D
1/2
=I
D
1/2
·A·D
1/2
(1.19a)
with elements
8
>
if i = j and deg(vi ) 6= 0
<1, p
Lij =
1/ deg(vi ) deg(vj ), if i 6= j and vi and vj are connected by edge (1.19b)
>
:
0,
otherwise
One can write L(G) as, cf. Eq. (1.16),
>
~
~
L(G) = B · B
~ is an |V | ⇥ |E|-matrix where
where B
8
p
>
< 1/pdeg(vi ), if edge es departs from vi
~ is = +1/ deg(vi ), if edge es arrives at vi
B
>
:
0,
otherwise
(1.20a)
(1.20b)
A ‘0-chain’ is a real-valued vertex function g : V ! R, and a ‘1-chain’ is a real-valued
~ = (B
~ is ) can be viewed as boundary operator that maps
edge function E ! R. Then B
>
~
~ si ) is a co-boundary operator
1-chains onto 0-chains, while the transposed matrix B = (B
that maps 0-chains onto 1-chains. Accordingly L can be viewed as an operator that maps
vertex functions g, which can be viewed as |V |-dimensional column vector, onto another
>
if i = j and deg(vi ) 6= 0
<1, p
Lij =
1/ deg(vi ) deg(vj ), if i 6= j and vi and vj are connected by edge (1.19b)
>
:
0,
otherwise
0-chain
0.4
That is, from one set of values ass
1-chain
One can write L(G) as, cf. Eq. (1.16),
-0.2
>
~
~
L(G) = B · B
(1.20a)
~ is an |V | ⇥ |E|-matrix where
where B
8
p0.1
0.1
>
< 1/pdeg(vi ), if edge es departs from vi
~ is = +1/ deg(vi ), if edge es arrives at vi
B
>
:
0,
otherwise
(1.20b)
A ‘0-chain’ is a real-valued vertex function gFigure
: V ! R,
and(a)
a ‘1-chain’
is a real-valued
6:
An
example
of 1-chain
~
~
edge function E ! R. Then B = (Bis ) can be viewed as boundary operator that maps
simplex);
example of 1-ch
~ > = ((b)
~ si )ais second
1-chains onto 0-chains, while the transposed matrix B
B
a co-boundary operator
that maps 0-chains onto 1-chains. Accordingly L can be viewed as an operator that maps
vertex functions g, which can be viewed as |V |-dimensional column vector, onto another
vertex function L · g, such that
"
#
X
1
g(vi )
g(vj )
p
p
(L · g)(vi ) = p
(1.21)
deg(vj )
deg(vi ) vj ⇠vi
deg(vi )
plex, one can deduce another set of
boundaries of each simplex by the
operation is very natural, and can
explained next.
where vj ⇠ vi denotes the set of adjacent nodes.
3.2.4
Implementation of the
where vj ⇠ vi denotes the set of adjacent nodes.
We denote the eigenvalues of L by
0=
0

1
 ... 
(6.22)
|V | 1
Abbreviating n = |V |, one can show that
P
(i)
i i  n with equality i↵ G has no isolated vertices.
(ii)
1
 n/(n
(iii) If n
1) with equality i↵ G is the complete graph on n
2 and G has no isolated vertices, then
(iv) If G is not complete, then
(v) If G is connected, then
(vi) If
i
= 0 and
i+1
1
n 1
n/(n
2 vertices.
1).
 1.
1
> 0.
> 0, then G has exactly i + 1 connected components.
(vii) For all i  n 1, we have
bipartite and nontrivial.
i
 2, with
n 1
= 2 i↵ a connected component of G is
(viii) The spectrum of a graph is the union of the spectra of its connected components.
See Chapter 1 in [Chu97] for proofs.
Examples:
• For a complete graph Kn on n
n/(n 1) (multiplicity n 1)
2 vertices, the eigenvalues are 0 (multiplicity 1) and
Examples:
• For a complete graph Kn on n
n/(n 1) (multiplicity n 1)
2 vertices, the eigenvalues are 0 (multiplicity 1) and
• For a complete bipartite graph Km,n on m + n vertices, the eigenvalues are 0 and 1
(multiplicity m + n 2) and 2.
• For the star Sn on n
and 2.
2 vertices, the eigenvalues are 0 and 1 (multiplicity n
• For the path Pn on n
k = 0, . . . , n 1.
2 vertices, the eigenvalues are
• For the cycle Cn on n
k = 0, . . . , n 1.
k
2 vertices, the eigenvalues are
n
• For
the
n-cube
Q
on
2
vertices, the eigenvalues are
n
✓ ◆
n
for k = 0, . . . , n.
k
9
k
=1
k
cos[⇡k/(n
= 1
2)
1)] for
cos[2⇡k/n] for
= 2k/n, with multiplicity
3.1
Graph Laplacian
Laplacian
e |V | ⇥ |V |-Laplacian matrix L(G) of a graph G, often also referred to as Kirchho↵
trix, is defined as the di↵erence between degree matrix and adjacency matrix
nce
L=D
A
(1.15a)
8
>
<deg(vi ), if i = j
Lij =
1,
if vi and vj are connected by edge
>
:
0,
otherwise
degree matrix
(1.15b)
we shall see below, this matrix provides an important characterization of the underlying
ph.
The |V | ⇥ |V |-Laplacian matrix can also be expressed in terms of the directed incidence
~ as
trix C,
>
~
~
L=C ·C
~ ir C
~ jr
Lij = C
,
For the graph in Fig. 1.3a, one finds
0
3
1
B 1 2
B
L=B
B 1 0
@ 1 0
0
1
1
0
2
1
0
1
0
1
3
1
1
0
1C
C
0C
C
1A
2
(1.16)
adjacency matrix
(1.17)
operties We denote the eigenvalues of L by
0

1
 ... 
|V |
e following properties hold:
) L is symmetric.
) L is positive-semidefinite, that is
i
0 for all i.
2
(1.18)
Laplacian matrix
0
B
B
L=B
B
@
3
1
1
1
0
1
2
0
0
1
1
0
2
1
0
1
0
1
3
1
1
0
1C
C
0C
C
1A
2
(1.17)
Properties We denote the eigenvalues of L by
0

1
 ... 
|V |
(1.18)
The following properties hold:
(i) L is symmetric.
(ii) L is positive-semidefinite, that is
i
0 for all i.
(iii) Every row sum and column sum of L is zero.2
(iv)
0
= 0 as the vector v 0 = (1, 1, . . . , 1) satisfies L · v 0 = 0.
(v) The multiplicity of the eigenvalue 0 of the Laplacian equals the number of connected
components in the graph.
(vi) The smallest non-zero eigenvalue of L is called the spectral gap.
(vii) For a graph with multiple connected components, L can written as a block diagonal
matrix, where each block is the respective Laplacian matrix for each component.
2
The degree of the vertex is summed with a -1 for each neighbor
7
Given a graph G, its line graph or derivative L[G] is a graph such that (i) each verte
of L[G] represents an edge of G and (ii) two vertices of L[G] are adjacent if and only
their corresponding edges share a common endpoint (‘are incident’) in G (Fig. ??). Th
Line graphs
of undirected
construction can be iterated
to obtain higher-order
line (orgraphs
derivative) graphs.
1.3
1. draw vertex for
each incidence
edge in G
Adjacency
and
2.
connect vertices if edges have joint point
Adjacency matrix Two vertices v1 and v2 of a graph are called adjacent, if they ar
connected by an edge. The adjacency matrix A(G) = (Aij ) is a |V | ⇥ |V |-matrix that list
all the connections in a graph. If the graph is simple, then A is symmetric and has onl
(a)
(b)
(c)
(d)
Figure 1.3: Construction of a line graph. These figures show a graph (a, with blue vertices
and its line graph (d, with green vertices). Each vertex of the line graph is shown labele
with the pair of endpoints of the corresponding edge in the original graph. For instanc
the green vertex on the right labeled 1,3 corresponds to the edge on the left between th
blue vertices 1 and 3. Green vertex 1,3 is adjacent to three other green vertices: 1,4 and 1,
Line graphs of directed graphs
Adjacency matrix Two vertices v1 and v2 of a graph are called adjacent, if they are
connected
edge.
Thematrix
adjacency
matrix matrix
A(G) C=of(A
a a|V
|V || ⇥
⇥|E|-matrix
|V |-matrix
Incidence
The incidence
graph
G is
withthat
Cis =lists
1
ij ) is
uch
that by
(i)aneach
vertex
Incidence matrix The incidence matrix C of graph G is a|V | ⇥ |E|-matrix with Cis = 1
if edge
is contained
in edge
es , and
C
For
the
graph in Fig.
1.3a,
with
is = 0 otherwise.
all the connections
in aifviedge
graph.
If
the
graph
is
simple,
then
A
is
symmetric
and
has
v is contained in edge e , and C = 0 otherwise. For the graph in only
Fig. 1.3a, with
i
s we have
is
adjacent if and
if
i = 1,only
. . . , 5 vertices
and s = 1, . . . , 6 edges,
i = 1, . . . , 5 vertices and s 0
= 1, . . . , 6 edges, 1
we have
nt’) in G (Fig. ??). This
1 1 1 00 0 0
1
B1 0 0 11 01 01C 0 0 0
B
C
tive) graphs.
B
B
C = 0 1 0 B01 10 00C 1 0 0C
C
(1.9)
B
C
@0 C0 =1 B00 11 10A 0 1 0C
B
C
@
0 0 0 10 00 11 0 1 1A
(1.9)
0 0 0 1 0 1
(a)
The incidence
(b) matrix C(G) of a graph
(c) G and the adjacency matrix
(d) A(L[G]) of its line
graph L[G]The
are related
by matrix C(G) of a graph G and the adjacency matrix A(L[G]) of its line
incidence
graph
L[G]=are
related
by 2I
>
A(L[G])
C(G)
· C(G)
,
A(L[G]) = C C
2
(1.10)
led
adjacent,
if
they
are
Figure 1.3: Construction of a line graph. These
figures show a graph (a, with blue vertices)
>
A(L[G])
= C(G)
· find
C(G) 2I
,
A(L[G]) = Cir Cis 2 rs
the
example
in Fig.
1.3,
we thus
its |-matrix
line graphFor
(d,
with
green
vertices).
Each
vertex of the line graph rsis shown
labeled
|and
⇥ |V
that
lists
0
1
with
the
pair
of
endpoints
of
the
corresponding
edge
in
the
original
graph. For instance,
the example in Fig. 1.3, we0thus
1 1find
1 0 0
symmetric and hasForonly
B1 0 1 0 1 0 C
rs
ir
is
rs
the green vertex on the right labeled 1,3 corresponds
to
on the
0 the edge
1 left between the
B
C
B1 1 0 00 11 1C
1 1 0 0
C
blue vertices 1 and 3. Green vertex 1,3 A(L[G])
is adjacent
to
three
other
green
vertices:
1,4 and
=B
(1.11)1,2
C
B1 0 0 B01 00 1C
1
0
1
0
C
B graph)
C
(corresponding to edges sharing the endpoint 1B
and
4,3 (corresponding
@in
B01 01 1A
0 the
1 1 blue
0 0 1 1C
B
C
B11 10 0and
C
0 0 =1 Image
to an edge sharing the endpoint 3 in the blueA(L[G])
graph).
wiki.
0 0 text
0 1source:
B
C
yielding the characteristic polynomial
p(L[G]; x) = (x + 2) x2 + x
yielding the characteristic polynomial
(d)
@0 1 1 0 0 1 A
0 0 1 1 1 0
1 [(x
3)x2
x + 2]
(1.10)
(1.11)
(1.12)
Directed incidence matrix In addition to the2 undirected incidence2 matrix C, we still
p(L[G]; x)~ = (x + 2) x + x 1 [(x 3)x
x + 2]
(1.12)
define a directed |V | ⇥ |E|-matrix C
as follows
8 4
Directed incidence>
In eaddition
thevundirected
incidence matrix C, we still
1, if edge
from
s departsto
i
<matrix
~ as follows
~ is|V=| ⇥+1,
define a directed
|E|-matrix
C
(1.13)
if edge C
es arrives at vi
>
:
8
Isomorphic graphs
image source: wiki
f(a) = 1 !
f(b) = 6!
f(c) = 8!
f(d) = 3!
f(g) = 5!
f(h) = 2!
f(i) = 4!
Whitney graph isomorphism theorem: Two connected graphs are isomorphic if and only if their !
line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the !
complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph.
Whitney, Hassler (January 1932). "Congruent Graphs and the Connectivity of Graphs". Amer. J. Mathematics (The Johns Hopkins University Press) 54 (1): 150–168
Line graphs of line graphs of ….
van Rooij & Wilf (1965):!
When G is a finite connected graph, only four possible behaviors are
possible for this sequence:!
! •! If G is a cycle graph then L(G) and each subsequent graph in this
sequence is isomorphic to G itself. These are the only connected
graphs for which L(G) is isomorphic to G.!
! •! If G is a claw K1,3, then L(G) and all subsequent graphs in the
sequence are triangles.!
! •! If G is a path graph then each subsequent graph in the sequence is
a shorter path until eventually the sequence terminates with an
empty graph.!
! •! In all remaining cases, the sizes of the graphs in this sequence
eventually increase without bound.!
!
If G is not connected, this classification applies separately to each
component of G.
Chromatic number
smallest number of colors needed to color the vertices of so that
no two adjacent vertices share the same color
NP complete problem: NP (verifiable in nondeterministic polynomial time) and NP-hard (any NP-problem can be translated into this problem)