## What's hot

Euler paths and circuits
Euler paths and circuits
03446940736

Matrix Representation Of Graph
Matrix Representation Of Graph
Abhishek Pachisia

Graph theory
Graph theory
iranian translate

Graph isomorphism
Graph isomorphism
Core Condor

Graph theory introduction - Samy
Graph theory introduction - Samy
Mark Arokiasamy

Graph Basic In Data structure
Graph Basic In Data structure
Ikhlas Rahman

Graph theory and life
Graph theory and life
Milan Joshi

KARNAUGH MAP(K-MAP)
KARNAUGH MAP(K-MAP)
mihir jain

Introduction to Graph Theory
Introduction to Graph Theory
Premsankar Chakkingal

Graph Theory: Cut-Set and Cut-Vertices
Graph Theory: Cut-Set and Cut-Vertices
Ashikur Rahman

Matrix representation of graph
Matrix representation of graph
Rounak Biswas

introduction to graph theory
introduction to graph theory
Chuckie Balbuena

Graph theory
Graph theory
Thirunavukarasu Mani

Ppt of graph theory
Ppt of graph theory
ArvindBorge

Graph Theory
Graph Theory
Rashmi Bhat

Multi ways trees
Multi ways trees
SHEETAL WAGHMARE

Graph Theory: Planarity & Dual Graph
Graph Theory: Planarity & Dual Graph
Ashikur Rahman

Karnaugh map
Karnaugh map
Vanitha Chandru

CS6702 Unit III coloring ppt
CS6702 Unit III coloring ppt
Abilaasha Ganesan

Graph theory
Graph theory
Manash Kumar Mondal

### What's hot(20)

Euler paths and circuits
Euler paths and circuits

Matrix Representation Of Graph
Matrix Representation Of Graph

Graph theory
Graph theory

Graph isomorphism
Graph isomorphism

Graph theory introduction - Samy
Graph theory introduction - Samy

Graph Basic In Data structure
Graph Basic In Data structure

Graph theory and life
Graph theory and life

KARNAUGH MAP(K-MAP)
KARNAUGH MAP(K-MAP)

Introduction to Graph Theory
Introduction to Graph Theory

Graph Theory: Cut-Set and Cut-Vertices
Graph Theory: Cut-Set and Cut-Vertices

Matrix representation of graph
Matrix representation of graph

introduction to graph theory
introduction to graph theory

Graph theory
Graph theory

Ppt of graph theory
Ppt of graph theory

Graph Theory
Graph Theory

Multi ways trees
Multi ways trees

Graph Theory: Planarity & Dual Graph
Graph Theory: Planarity & Dual Graph

Karnaugh map
Karnaugh map

CS6702 Unit III coloring ppt
CS6702 Unit III coloring ppt

Graph theory
Graph theory

## Similar to Sub matrices - Circuit Matrix

Elements of Graph Theory for IS.pptx
Elements of Graph Theory for IS.pptx
miki304759

Graphs
Graphs
amudha arul

ch10.5.pptx
ch10.5.pptx
GauravGautam216125

Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring
Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring
Saurabh Kaushik

Linear Algebra Gauss Jordan elimination.pptx
Linear Algebra Gauss Jordan elimination.pptx
Maths Assignment Help

Introduction to-graph-theory-1204617648178088-2
Introduction to-graph-theory-1204617648178088-2
Houw Liong The

graph theory
graph theory
Shashank Singh

09_DS_MCA_Graphs.pdf
09_DS_MCA_Graphs.pdf
Prasanna David

Graphs.pdf
Graphs.pdf
pubggaming58982

Network Topology
Network Topology
Harsh Soni

Graphs (Models & Terminology)
Graphs (Models & Terminology)
zunaira saleem

Network Topology.PDF
Network Topology.PDF
parameshwar7

Algorithms and data Chapter 3 V Graph.pptx
Algorithms and data Chapter 3 V Graph.pptx
zerihunnana

PPT on Graph Theory ( Tree, Cotree, nodes, branches, incidence , tie set and ...
PPT on Graph Theory ( Tree, Cotree, nodes, branches, incidence , tie set and ...
SUTAPAMUKHERJEE12

Ass. (3)graph d.m
Ass. (3)graph d.m
Syed Umair

Chapter 1
Chapter 1
MeeraMeghpara

Ch08.ppt
Unit 2: All
Unit 2: All
Hector Zenil

Class01_Computer_Contest_Level_3_Notes_Sep_07 - Copy.pdf
Class01_Computer_Contest_Level_3_Notes_Sep_07 - Copy.pdf
ChristianKapsales1

Bba i-bm-u-2- matrix -
Bba i-bm-u-2- matrix -
Rai University

### Similar to Sub matrices - Circuit Matrix(20)

Elements of Graph Theory for IS.pptx
Elements of Graph Theory for IS.pptx

Graphs
Graphs

ch10.5.pptx
ch10.5.pptx

Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring
Graph Theory,Graph Terminologies,Planar Graph & Graph Colouring

Linear Algebra Gauss Jordan elimination.pptx
Linear Algebra Gauss Jordan elimination.pptx

Introduction to-graph-theory-1204617648178088-2
Introduction to-graph-theory-1204617648178088-2

graph theory
graph theory

09_DS_MCA_Graphs.pdf
09_DS_MCA_Graphs.pdf

Graphs.pdf
Graphs.pdf

Network Topology
Network Topology

Graphs (Models & Terminology)
Graphs (Models & Terminology)

Network Topology.PDF
Network Topology.PDF

Algorithms and data Chapter 3 V Graph.pptx
Algorithms and data Chapter 3 V Graph.pptx

PPT on Graph Theory ( Tree, Cotree, nodes, branches, incidence , tie set and ...
PPT on Graph Theory ( Tree, Cotree, nodes, branches, incidence , tie set and ...

Ass. (3)graph d.m
Ass. (3)graph d.m

Chapter 1
Chapter 1

Ch08.ppt
Ch08.ppt

Unit 2: All
Unit 2: All

Class01_Computer_Contest_Level_3_Notes_Sep_07 - Copy.pdf
Class01_Computer_Contest_Level_3_Notes_Sep_07 - Copy.pdf

Bba i-bm-u-2- matrix -
Bba i-bm-u-2- matrix -

BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
MIGUELANGEL966976

insn4465

Heat Resistant Concrete Presentation ppt
Heat Resistant Concrete Presentation ppt
mamunhossenbd75

Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf

The Python for beginners. This is an advance computer language.
The Python for beginners. This is an advance computer language.
sachin chaurasia

2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
Yasser Mahgoub

New techniques for characterising damage in rock slopes.pdf
New techniques for characterising damage in rock slopes.pdf
wisnuprabawa3

Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Christina Lin

Textile Chemical Processing and Dyeing.pdf
Textile Chemical Processing and Dyeing.pdf
NazakatAliKhoso2

Engine Lubrication performance System.pdf
Engine Lubrication performance System.pdf
mamamaam477

Advanced control scheme of doubly fed induction generator for wind turbine us...
Advanced control scheme of doubly fed induction generator for wind turbine us...
IJECEIAES

A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
IJNSA Journal

Literature Review Basics and Understanding Reference Management.pptx
Literature Review Basics and Understanding Reference Management.pptx
Dr Ramhari Poudyal

Modelagem de um CSTR com reação endotermica.pdf
Modelagem de um CSTR com reação endotermica.pdf
camseq

ACEP Magazine edition 4th launched on 05.06.2024
ACEP Magazine edition 4th launched on 05.06.2024
Rahul

Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
IJECEIAES

Casting-Defect-inSlab continuous casting.pdf
Casting-Defect-inSlab continuous casting.pdf

CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
jpsjournal1

Computational Engineering IITH Presentation
Computational Engineering IITH Presentation
co23btech11018

Sinan KOZAK

BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf

Heat Resistant Concrete Presentation ppt
Heat Resistant Concrete Presentation ppt

Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf

The Python for beginners. This is an advance computer language.
The Python for beginners. This is an advance computer language.

2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf

New techniques for characterising damage in rock slopes.pdf
New techniques for characterising damage in rock slopes.pdf

Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines

Textile Chemical Processing and Dyeing.pdf
Textile Chemical Processing and Dyeing.pdf

Engine Lubrication performance System.pdf
Engine Lubrication performance System.pdf

Advanced control scheme of doubly fed induction generator for wind turbine us...
Advanced control scheme of doubly fed induction generator for wind turbine us...

A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS

Literature Review Basics and Understanding Reference Management.pptx
Literature Review Basics and Understanding Reference Management.pptx

Modelagem de um CSTR com reação endotermica.pdf
Modelagem de um CSTR com reação endotermica.pdf

ACEP Magazine edition 4th launched on 05.06.2024
ACEP Magazine edition 4th launched on 05.06.2024

Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...

Casting-Defect-inSlab continuous casting.pdf
Casting-Defect-inSlab continuous casting.pdf

CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT

Computational Engineering IITH Presentation
Computational Engineering IITH Presentation

### 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.
Current LanguageEnglish
Español
Portugues
Français
Deutsche