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SUB MATRICES – CIRCUIT MATRIX
(GROUP – 2)
MAT2002 – GROUP ACTIVITY
GROUP MEMBERS
ADITI AGRAWAL
20BHI10050
MANAN LADDHA
20BCY10115
RUSHIL BHATNAGAR
20BAI10299
K K ISHVINTHA SREE
20BHI10029
AAKANKSHA PRIYA
20BCY10030
D
D
Subgraph: A graph G’ is said to be a subgraph of a graph G, if all the edges of G’ are in G, and each
edge of G’ has the same end vertices in G’ as in G.
Let there be a graph G, and let G’ be its subgraph. Let A(G’) and A(G) be the incidence matrices of
G’ and G, respectively. Then, A(G’) is a submatrix of A(G). There is a one-to-one correspondence be
tween each n by k submatrix of A(G) and a subgraph of G with k edges, k being any positive intege
r less than e and n being the number of vertices in G.
Submatricesof A(G)exhibitspecialproperties
Theorem:Let A(G)be an incidencematrixof a connectedgraphG withn vertices. An (n-1)by (n-1)submatrixof A(G)is nonsi
ngularif and only if then-1 edgescorrespondingto then-1 columnsof thismatrixconstitutea spanningtreein G.
Proof:Everysquare submatrixof ordern-1 in A(G)is thereducedincidencematrixof thesame subgraphin G withn-1 edges,
andviceversa. We knowthata square submatrixof A(G)is nonsingularif and onlyif thecorresponding subgraphis a tree. T
he treein thiscaseis a spanningtree, because it containsn-1 edgesof then-vertexgraph.
Hence, an (n-1)by (n-1)submatrixof an incidence matrixis nonsingularif and only if then-1 edgescorrespondingto then-1 c
olumnsof thismatrixconstitutea spanningtreein G.
Walk & Circuit
Walk: A walk is a sequence of vertices a
nd edges of a graph i.e. if we traverse a
graph then we get a walk. Repetition of
vertex and edges is allowed in a walk.
.
Circuit: A walk that starts and ends at a
same vertex and contains no repeated e
dges. Note: A circuit is a closed walk, bu
t a closed walk need not be a circuit.
Closed walk: If a walk starts and ends
at the same vertex, then it is said to b
e a closed walk.
For example, in the figure:
Walk: A – B – C – E – A – D , E – D –
E – F, etc.
Circuit: A – D – E – A , A – B – C – F –
E – D – A, etc.
Circuit Matrix
Let the graph G have m edges(e) and let q be the number of different cycles(C) in G. Then the circ
uit matrix B = [bij] of G is a (0, 1) − matrix of order q × m, with
Forexample:The graph G1 has four different cycles Z1 = {e1, e2}, Z2 = {e3, e5, e7}, Z3 = {e4, e6, e7}
and Z4 = {e3, e4, e6, e5}. Its circuitmatrix = B(G1)
Properties of Circuit Matrix
1 2 3 4 5
A column of all zeros c
orresponds to an edge
which does not belong
to any cycle.
Each row of B(G) is
a circuit vector.
The number of ones
in a row is equal to t
he number of edges
in the corresponding
cycle.
A circuit matrix has t
he property of repre
senting a self-loop b
y having a single 1 i
n the corresponding
row.
Permutation of any t
wo rows or columns
in a cycle matrix cor
responds to relabeli
ng the cycles and th
e edges.
Theorem 7.4
5. If the graph G is separable (or disconnected) and consists of two blocks (or components) H1 and
H2, then the cycle matrix B(G) can be written in a block-diagonal form as:
,where B(H1) and B(H2) are the cycle matrices of H1 and H2.
STATEMENT
• Let B and A be, respectively, the circuit matrix and the incidence
matrix (of a self-loop-free graph) whose columns are arranged usi
ng the same order of edges. Then every row of B is orthogonal to
every row A; that is,
A.BT = B.AT = 0
• where the superscripted T denotes a transposed matrix.
PROOF
Consider a vertex v and a circuit ᴦ in the graph G. either v is in ᴦ or it is not. If v is in ᴦ, there
’s no edge in the circuit ᴦ that is incident on v. on the other hand if v is in v, the number of e
dges of ᴦ incident on v is exactly two.
With this remark in mind, consider the i-th row in and the j-th row in B. Since the edges are
arranged in the same order, the non zero entries are in corresponding positions occur only if
the particular edge is incident on the i-th vertex and is also in the j-th circuit.
If the i-th vertex is not in the circuit, there is no such nonzero entry, and the dot product of
the two rows is zero. If the i-th vertex is in the j-th circuit, there will be exactly two 1’s in t
he sum of the products of the individual entries. Since 1+1=0, the product of the two arbitrar
y rows – one from A and B and other from B – is zero. Hence the theorem is proved. An Exa
mple:
.
0 0 0 1 0 1 0 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 1
1 1 1 0 1 0 0 0
0 0 1 1 0 0 1 0
1 1 0 0 0 0 0 0
1 0 0 0
1 0 0 0
0 1 0 1
0 0 1 1
0 1 0 1
0 0 1 1
0 1 1 0
0 0 0 0
= [O] = 0
Fundamental Circuit Matrix and Rank of B
• A Submatrix, of a circuit matrix in which all rows correspond to a set of funda
mental circuits is called a fundamental circuit matrix Bf. As in matrices A and
B, permutations of rows or columns do not affect Bf. If n is the number of ver
tices and e is the number of edges in a connected graph, then Bf is (e-n+1) b
y e matrix, because the number of fundamental circuits is e-n+1, each fundam
ental circuit being produced by one chord.
• Let us arrange the columns in Bf such that all the e-n+1 chords correspond t
o the first e-n+1 columns. Further let us rearrange the rows such that the firs
t row corresponds to the fundamental circuit made by the chord in the first co
lumn, the second row to the fundamental circuit made by the second and so o
n. This indeed is how the fundamental circuit matrix is arranged in the below fi
gure (b)
e1 e3 e6 e1 e4 e5 e7
1 0 0 1 1 0 1
0 1 0 0 1 0 1
0 0 1 0 0 1 1
e3
e4
e5
e6
Graph and its
fundamental circuit
matrix (with respect
to the spanning tree
shown in blue
colour)
(a)
(b)
A matrix Bf thus arranged can be written as
Bf=[Iµ | Bf] - - - - (1)
Where, Iµ is an identity matrix of order µ = e-n+1 and Bf is the
remaining µ by (n-1) submatrix, corresponding to the branches
of the spanning tree.
From equation (1) it is clear that the
Rank of the Bf = µ = e-n+1
Since Bf is a submatrix of the circuit matrix B, the
Rank of B>= e-n+1.
THANK YOU !

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Sub matrices - Circuit Matrix

  • 1. SUB MATRICES – CIRCUIT MATRIX (GROUP – 2) MAT2002 – GROUP ACTIVITY
  • 2. GROUP MEMBERS ADITI AGRAWAL 20BHI10050 MANAN LADDHA 20BCY10115 RUSHIL BHATNAGAR 20BAI10299 K K ISHVINTHA SREE 20BHI10029 AAKANKSHA PRIYA 20BCY10030 D D
  • 3. Subgraph: A graph G’ is said to be a subgraph of a graph G, if all the edges of G’ are in G, and each edge of G’ has the same end vertices in G’ as in G. Let there be a graph G, and let G’ be its subgraph. Let A(G’) and A(G) be the incidence matrices of G’ and G, respectively. Then, A(G’) is a submatrix of A(G). There is a one-to-one correspondence be tween each n by k submatrix of A(G) and a subgraph of G with k edges, k being any positive intege r less than e and n being the number of vertices in G.
  • 4. Submatricesof A(G)exhibitspecialproperties Theorem:Let A(G)be an incidencematrixof a connectedgraphG withn vertices. An (n-1)by (n-1)submatrixof A(G)is nonsi ngularif and only if then-1 edgescorrespondingto then-1 columnsof thismatrixconstitutea spanningtreein G. Proof:Everysquare submatrixof ordern-1 in A(G)is thereducedincidencematrixof thesame subgraphin G withn-1 edges, andviceversa. We knowthata square submatrixof A(G)is nonsingularif and onlyif thecorresponding subgraphis a tree. T he treein thiscaseis a spanningtree, because it containsn-1 edgesof then-vertexgraph. Hence, an (n-1)by (n-1)submatrixof an incidence matrixis nonsingularif and only if then-1 edgescorrespondingto then-1 c olumnsof thismatrixconstitutea spanningtreein G.
  • 5. Walk & Circuit Walk: A walk is a sequence of vertices a nd edges of a graph i.e. if we traverse a graph then we get a walk. Repetition of vertex and edges is allowed in a walk. . Circuit: A walk that starts and ends at a same vertex and contains no repeated e dges. Note: A circuit is a closed walk, bu t a closed walk need not be a circuit. Closed walk: If a walk starts and ends at the same vertex, then it is said to b e a closed walk. For example, in the figure: Walk: A – B – C – E – A – D , E – D – E – F, etc. Circuit: A – D – E – A , A – B – C – F – E – D – A, etc.
  • 6. Circuit Matrix Let the graph G have m edges(e) and let q be the number of different cycles(C) in G. Then the circ uit matrix B = [bij] of G is a (0, 1) − matrix of order q × m, with Forexample:The graph G1 has four different cycles Z1 = {e1, e2}, Z2 = {e3, e5, e7}, Z3 = {e4, e6, e7} and Z4 = {e3, e4, e6, e5}. Its circuitmatrix = B(G1)
  • 7. Properties of Circuit Matrix 1 2 3 4 5 A column of all zeros c orresponds to an edge which does not belong to any cycle. Each row of B(G) is a circuit vector. The number of ones in a row is equal to t he number of edges in the corresponding cycle. A circuit matrix has t he property of repre senting a self-loop b y having a single 1 i n the corresponding row. Permutation of any t wo rows or columns in a cycle matrix cor responds to relabeli ng the cycles and th e edges.
  • 8. Theorem 7.4 5. If the graph G is separable (or disconnected) and consists of two blocks (or components) H1 and H2, then the cycle matrix B(G) can be written in a block-diagonal form as: ,where B(H1) and B(H2) are the cycle matrices of H1 and H2. STATEMENT • Let B and A be, respectively, the circuit matrix and the incidence matrix (of a self-loop-free graph) whose columns are arranged usi ng the same order of edges. Then every row of B is orthogonal to every row A; that is, A.BT = B.AT = 0 • where the superscripted T denotes a transposed matrix.
  • 9. PROOF Consider a vertex v and a circuit ᴦ in the graph G. either v is in ᴦ or it is not. If v is in ᴦ, there ’s no edge in the circuit ᴦ that is incident on v. on the other hand if v is in v, the number of e dges of ᴦ incident on v is exactly two. With this remark in mind, consider the i-th row in and the j-th row in B. Since the edges are arranged in the same order, the non zero entries are in corresponding positions occur only if the particular edge is incident on the i-th vertex and is also in the j-th circuit. If the i-th vertex is not in the circuit, there is no such nonzero entry, and the dot product of the two rows is zero. If the i-th vertex is in the j-th circuit, there will be exactly two 1’s in t he sum of the products of the individual entries. Since 1+1=0, the product of the two arbitrar y rows – one from A and B and other from B – is zero. Hence the theorem is proved. An Exa mple: . 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0 0 = [O] = 0
  • 10. Fundamental Circuit Matrix and Rank of B • A Submatrix, of a circuit matrix in which all rows correspond to a set of funda mental circuits is called a fundamental circuit matrix Bf. As in matrices A and B, permutations of rows or columns do not affect Bf. If n is the number of ver tices and e is the number of edges in a connected graph, then Bf is (e-n+1) b y e matrix, because the number of fundamental circuits is e-n+1, each fundam ental circuit being produced by one chord. • Let us arrange the columns in Bf such that all the e-n+1 chords correspond t o the first e-n+1 columns. Further let us rearrange the rows such that the firs t row corresponds to the fundamental circuit made by the chord in the first co lumn, the second row to the fundamental circuit made by the second and so o n. This indeed is how the fundamental circuit matrix is arranged in the below fi gure (b)
  • 11. e1 e3 e6 e1 e4 e5 e7 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 e3 e4 e5 e6 Graph and its fundamental circuit matrix (with respect to the spanning tree shown in blue colour) (a) (b)
  • 12. A matrix Bf thus arranged can be written as Bf=[Iµ | Bf] - - - - (1) Where, Iµ is an identity matrix of order µ = e-n+1 and Bf is the remaining µ by (n-1) submatrix, corresponding to the branches of the spanning tree. From equation (1) it is clear that the Rank of the Bf = µ = e-n+1 Since Bf is a submatrix of the circuit matrix B, the Rank of B>= e-n+1.