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