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19306 Graph Theory1
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Transcript of 19306 Graph Theory1
Graph Theory
Lecture 27: Graph Theory in Circuit Analysis
Replace all branches by a line - Graph
Directed and Un-directed Graph
Node and branch are incident if the node is terminal to the branch
Number of branches incident at a node – degree of node
Connected Graph – path between every pair of nodes
Planar and Non-Planar Graphs
Tree and Co-Tree
A tree is a connected sub-graph with all the nodes but no closed paths/loops
Branches of a tree – Twigs
Generally, n nodes in a tree – (n-1) branches
Remaining branches of Graph excluding any tree are called Links and the set of links is called Co-Tree
B branches and n nodes in Graph – (n-1) twigs and (b-n+1) links
Number of twigs – rank of a Tree
Incidence Matrix
The incidence of elements/branches to nodes in a connected graph – Node Incidence Matrix
Arrows – direction of current flow or Voltage rise
n X b – number of nodes X number of branches
Aij = 1, if j th branch is incident to and oriented away from i th node
Aij = -1, if j th branch is incident to and oriented towars from i th node
Aij = 1, if j th branch is not incident to i th node
Properties of Incident Matrix
Reduced Incidence Matrix A1 – (n-1) X b
Number of possible Trees – Det[A1*A1’ ]
Incidence Matrix and KCL
A1.I = 0 --- n-1 linearly independent equations
A1 – reduced incidence matrix
I = [i1; i2; …… ; in] – branch currents