EE 529
Circuit and Systems Analysis
Lecture 3
Mustafa Kemal Uyguroğlu
EASTERN MEDITERRANEAN UNIVERSITY
Basic Concepts of the Graph Theory
 Definition: In a connected graph G of v
vertices the subgraph T that satisfies the
following properties is called a tree.
T is connected
 T contains all the vertices of G
 T contains no circuit,
 T contains exactly v-1 number of edges.

In every connected graph G there exists at least
one tree.
Basic Concepts of the Graph Theory
 Let G have p separated parts G1, G2, ..., Gp,
that is G=G1G2...Gp, and let Ti be a tree
in Gi, i=1,2,...,p, then,T=T1 T2... Tp is
called a forest of G.
 DEFINITION: The complement of a tree is
called a co-tree and the complement of a
forest is called a co-forest. The edges of a
tree or a forest are called branches and the
edges of a co-tree or co-forest are called
chords (links).
Basic Concepts of the Graph Theory
v1
v4
e2
e4
v3
e6
e1
e5
e3
v5
v2 9 possible trees and corresponding co-trees:
T1  {e2 , e3 , e4 , e5 }
T4  {e1 , e2 , e5 , e6 } T7  {e2 , e3 , e5 , e6 }
T1  e1 , e6 
T4  e3 , e4 
T7  e1 , e4 
T2  {e1 , e2 , e4 , e6 }
T5  {e1 , e3 , e4 , e6 }
T8  {e1 , e2 , e4 , e5 }
T2  e3 , e5 
T5  e2 , e5 
T8  e3 , e6 
T3  {e1 , e3 , e5 , e6 }
T6  {e2 , e3 , e4 , e6 }
T9  {e1 , e3 , e4 , e5 }
T4  e2 , e4 
T6  e1 , e5 
T9  e2 , e6 
Basic Concepts of the Graph Theory
 DEFINITION: Let G be a graph and let r and
 be respectively the number of branches
and chords of G, then r and  are called
respectively the rank and the nullity of the
graph.
 THEOREM: Let G have v vertices, e edges
and p connected parts, then its rank and
nullity are given respectively by
r=v - p
and
=e – v + p
Basic Concepts of the Graph Theory
 DEFINITION: Let G be a connected graph
and let T and T’ be tree and co-tree
respectively, that is G=TT’. Let a chord
e’T’ and its unique tree path (a path which
is formed by the branches of T) define a
circuit. This circuit is called the fundamental
circuit (f-circuit) of G. All such circuits
defined by all the chords of T’ are called the
fundamental circuits (f-circuits) of G. If G is
not connected, then the f-circuits are defined
with respect to a forest.
Basic Concepts of the Graph Theory
 Note that the number of f-circuits is given by
the nullity of G and that, with respect to a
chosen tree T of G, each f-circuit contains
one and only one chord.
Basic Concepts of the Graph Theory
Consider the following graph
v1
v4
e2
e5
e4
v3
v5
e6
e1
e3
e8
e7
v6
v2
f-circuits:
cf1={e3,e1,e2}
cf2={e6,e8,e4,e5}
cf3={e7,e8,e4,e5}
Nullity of G
=e-v+p=8-6+1=3
Basic Concepts of the Graph Theory
 DEFINITION: The cut-set of a graph G is the
subgraph Gx of G consisting of the set of
edges satisfying the following properties:
 The
removal of Gx from G reduces the rank of
G exactly by one.
 No proper subgraph of Gx has this propery.
If G is connected then the first property in the
above definition can be replaced by the
following phrase.
 The removal of Gx from G separates the given
connected graph G into exactly two connected
subgraphs.
Basic Concepts of the Graph Theory
Consider the following graph and the following set of edges
v1
v4
e2
e5
e4
v3
v5
e6
e1
e3
e8
v6
v2
G1={e1,e2}
is also a cut-set
G2={e4,e6,e7}
Cut-set
G3={e2,e3,e4,e8}
G4={e2,e3,e6}
e7
is not a cut-set, because the removal of G3 from
G results in three connected subgraphs
is not a cut-set, because a subset of G4 is cut-set
Basic Concepts of the Graph Theory
 DEFINITION: Let G be a connected graph
and let T be its tree. The branch etT
defines a unique cut-set (a cut-set which is
formed by et and the chords of G). This cutset is called the fundamental cut-set (fcutset) of G. All such cut-sets defined by all
the branches of T are called the
fundamental cut-sets (f-cutsets) of G. If G is
not connected then the f-cutsets are defined
with respect to a forest.
Basic Concepts of the Graph Theory
 Note that the number of fundamental cut-
sets is given by the rank of G and with
respect to a chosen tree T of G, each
fundamental cut-set contains one and only
one branch.
Basic Concepts of the Graph Theory
Consider the following graph
v1
v4
e2
e5
e4
v3
v5
e6
e1
e8
e3
e7
v6
v2
f-cutsets:
xf1={e1,e3}
xf2={e2,e3}
xf3={e4,e6,e7}
xf4={e5,e6,e7}
xf5={e8,e6,e7}
Matrices of Oriented Graphs
v4
v1
e8
v8
e9
e5
v5
v2
e11
e10
v3
e3
v6
e4
e2
e1
e6
e7
v7
• The edge e1 which has an orientation from vertex v1 to
vertex v2 simply indicates that any transmission from v1 to v2
along e1 is assumed to be positive.
• Any transmission from v2 to v1 along e1 is assumed to be
negative.
Matrices of Oriented Graphs
DEFINITION: Let e and v represent
respectively the number of edges and
vertices of a graph G. The incident matrix
   ij 
v ,e
having v rows and e columns is defined as
1 if edge e j is incident at vertex vi and is oriented away from vi

 ij  1 if edge e j is incident at vertex vi and is oriented toward vi
0 if edge e is not incident on vertex v
j
i

Matrices of Oriented Graphs
v4
v1
e8
e4
e2
e1
v6
e6
e11
e10
v3
v8
e9
e5
e3
v5
v7
e7
v2
Incident Matrix:
 e1 

 1
 1

 0
 0

 0
 0

 0
 0

 e2   e3   e4   e5   e6   e7   e8   e9   e10   e11 
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
1
1
0
0

 v1  
 v2  

 v3  
 v4  
 v5  
 v6  
 v7  
 v8  
Property:
Any column of 
contains exactly two
nonzero entries of
opposite sign.
Matrices of Oriented Graphs
Property: The determinant of any square
submatrix of order q (1  q  v) of  is either
one of the following values: 1, -1, 0.
Now, consider a graph G of p connected
parts:
 E1 E2
 0
 1
2
  0


0
0

Ep
0
0
p

V1 
V2 


V p 
Matrices of Oriented Graphs
 DEFINITION: For a connected graph G, the
matrix , obtained by deleting any one of the
rows of the incidence matrix is called the
reduced incident matrix.
 Note that since any column of  contains
exactly two nonzero entries of opposite sign,
one can uniquely determine the incident
matrix when the reduced incident matrix is
given.
Matrices of Oriented Graphs
Property: Any square submatrix i of order
v-1 of the reduced incidence matrix  of G is
nonsingular if and only if the columns of i
correspond to the branches of a tree T of G.
Example
e2
v1
e1
v2
e3
v3
 e1 e2 e3

 1 1 0 v
1

 0 1 1 v2 


 1 0 1 v3 
Reduced incidence matrix
e1 e2 e3


   1 1 0 v1 
 0 1 1 v2 
i: nonsingular
Matrices of Oriented Graphs
In a graph G let k be the number of circuits
and let an arbitrary circuit orientation be
assigned to each one of these circuits.
DEFINITION: The circuit matrix
B  bij 
k ,e
for a graph G of e edges and k circuits is
defined as
1 if edge e j is in the circuit ci and the orientations of e j and ci are coincident

bij  1 if edge e j is in the circuit ci and the orientations of e j and ci are opposite
0 if edge e is not in the circuit c
j
i

Matrices of Oriented Graphs
Consider the following graph
v1
e3
e2
e1
e4
c2
c3 c1
c5
v4
c6
v2
e5
c4
e6
 e1 e2 e3 e4 e5 e6
 1 1 0 0 0 0

 0 1 1 1 0 0

B   1 0 1 1 0 0
 0 0 0 1 1 1

 1 0 1 0 1 1
 0 1 1 0 1 1

v3

c1 
c2 

c3 
c4 

c5 
c6 
Matrices of Oriented Graphs
Let bi represent the row of B that
corresponds to circuit ci. The circuits ci,...,cj
are independent if the rows bi,... bj are
independent.
DEFINITION: The f-circuit matrix Bf of a
graph G with respect to some tree T is
defined as the circuit matrix consisting of the
fundamental circuits of G only whose
orientations are chosen in the same direction
as that of defining chords.
Matrices of Oriented Graphs
The fundamental circuit matrix Bf of a graph
G with respect to some tree T can always be
written as
 (r )
Bf   B

( )

U  ( )

Matrices of Oriented Graphs
Consider the following graph
v1
e3
e2
v2
e4
e5
e1
e6
v4
 e1
 1
B f   1

 1

Bf  B
v3

0 0 1 0 0   e2 
1 0 0 1 0   e4 

1 1 0 0 1  e6 

U
e3
e5
e2
e4
e6
Matrices of Oriented Graphs

THEOREM: If the column orderings of the
circuit and incident matrices are identical then
B  0
T
or
B  0
T
Matrices of Oriented Graphs
Consider the following graph
v1
e3
e2
e1
e4
c2
c3 c1
c5
c6
v4
 e1 e2 e3 e4 e5 e6
 1 1 0 0 0 0

 0 1 1 1 0 0

B   1 0 1 1 0 0
 0 0 0 1 1 1

 1 0 1 0 1 1
 0 1 1 0 1 1

v2
e5
c4
e6

c1 
c2 

c3 
c4 

c5 
c6 
v3
 e1 e2 e3 e4 e5 e6  v
 1 1 1 0 0 0 1

v
   0 0 1 1 1 0  2
v
 0 0 0 0 1 1 3
 1 1 0 1 0 1 v4


Matlab Program..
>> b=[-1 1 0 0 0 0;0 -1 1 1 0 0;-1 0 1 1 0 0;
0 0 0 -1 1 1; -1 0 1 0 1 1;0 -1 1 0 1 1];
>> p=[1 1 1 0 0 0;0 0 -1 1 1 0; 0 0 0 0 -1 1; -1 -1 0 -1 0 -1];
>> b*p‘
ans =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
>> p*b’
ans =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
>>
Matrices of Oriented Graphs
In a graph G let x be the number of cut-sets
having arbitrary orientations. Then, we have
the following definition.
DEFINITION: The cut-set matrix
A  aij 
x ,e
for a graph G of e edges and x cut-sets is
defined as
1 if edge e j is in cut-set x i and the orientations of e j and x i are coincident

aij  1 if edge e j is in cut-set x i and the orientations of e j and x i are opposite
0 if edge e is not in cut-set x
j
i

Matrices of Oriented Graphs
Consider the following graph and its seven
possible cut-sets
v1
e5
e6
e4
 e1 e2 e3
 x1
1

1

1
0
0
0


 1 0 0 1 0 1 x2

x
 0 0 1 0 1 1 3
A   0 1 0 1 1 0  x4

x
 1 0 0 0 1 1 5
 1 0 1 1 1 0  x6


 0 1 1 1 0 0  x7
x1
x7
e2
x6
e3
e1
v0
e5
e4
x4
v2
x2
x5
e6
x3
v3
Matrices of Oriented Graphs
DEFINITION: The f-cutset matrix Af of a
graph G with respect to some tree T is
defined as the cut-set matrix consisting of the
fundamental cut-set of G only whose
orientations are chosen in the same direction
as that of defining branches.
Matrices of Oriented Graphs
The fundamental cut-set matrix Af of a
graph G with respect to some tree T can
always be written as
 (r )
Af  U

( )

A  (r )

Matrices of Oriented Graphs
Consider the following graph
v1
e2
e3
e1
v0
e5
e4
v2
e6
e3
e6
 e2 e4 e5 e1

1
0
0

1
1
0

 e2
A f   0 1 0 1 0 1 e4


0
0
1
0

1
1

 e5


v3
Matrices of Oriented Graphs

THEOREM: If the column orderings of the
circuit and cut-set matrices are identical then
AB  0
T
or
BA  0
T
Matrices of Oriented Graphs
Consider the following
e5
e6
e4
 e1 e2 e3
 x1
1

1

1
0
0
0


 1 0 0 1 0 1 x2

x
0
0
1
0

1

1

 3
A   0 1 0 1 1 0  x4

x
 1 0 0 0 1 1 5
 1 0 1 1 1 0  x6


 0 1 1 1 0 0  x7
graph
v1
e2
e3
e1
v0
e5
e4
v2
e6
v3
e3
e5
e2
e4
 e1
0 0
1 1
0
1
1 0 1 0

1
 0 1 1 0
B  1
1 0
0 1

1 0
 0 1 1
 1 0 1 1
1

 1 0 1 0
0

1  c1
0  c2

0  c3
1 c4
c
1 5
0  c6

1 c7
e6
Matrices of Oriented Graphs

THEOREM: In a graph G let the fundamental
circuit and cut-set matrices with respect to a
tree to be written as
Bf =  B U 
and
Af =  U A 
If the column orderings are identical then
A = B T
FUNDAMENTAL POSTULATES
Now, Let G be a connected graph having e
edges and let
xT   x1  t  , x2  t  ,
and
xe  t  
y T   y1  t  , y2  t  ,
ye  t  
be two vectors where xi and yi, i=1,...,e,
correspond to the across and through
variables associated with the edge i
respectively.
FUNDAMENTAL POSTULATES
2. POSTULATE Let B be the circuit matrix of the
graph G having e edges then we can write the
following algebraic equation for the across variables
of G
Bx = 0
3. POSTULATE Let A be the cut-set matrix of the
graph G having e edges then we can write the
following algebraic equation for the through
variables of G
Ay = 0
FUNDAMENTAL POSTULATES
2. POSTULATE is called the circuit equations of
electrical system. (is also referred to as Kirchoff’s
Voltage Law)
3. POSTULATE is called the cut-set equations of
electrical system. (is also referred to as Kirchoff’s
Current Law)
Fundamental Circuit & Cut-set Equations
 Consider a graph G and a tree T in G. Let the vectors
x and y partitioned as
xT   xTb
xTc 
y T   y Tb
y Tc 
 where xb (yb) and xc (yc) correspond to the across
(through) variables associated with the branches and
chords of the tree T, respectively.
 Then
 xb 
B U   x   0
 c
xc  Bxb
fundamental circuit
equation
and
yb 
U A  y  = 0
 c
y b = -Ay c
fundamental cut-set
equation
Example
 In the following figure, an oriented graph with
a tree, indicated in heavy lines, is given.



Obtain the f-circuit and f-cutset matrices
Obtain the reduced incident matrix, , (take
(a) as reference
Explain how you would derive the matrix A
from  and show this derivation on the 
matrix
2
b
c
1
a
3
6
5
4
d
Series & Parallel Edges
 Definition: Two edges ei and ek are said to be
connected in series if they have exactly one
common vertex of degree two.
v0
ei
ek
Series & Parallel Edges
 Definition: Two edges ei and ek are said to be
connected in parallel if they are incident at
the same pair of vertices vi and vk.
vi
ek
ei
vk
n+1 edges connected in series or in parallel?