# Incidence Matrix

By Mohit Uniyal|Updated : August 26th, 2022

The Incidence Matrix describes which branches are incident at which nodes and what are the orientations relative to the respective nodes. This incidence matrix is also termed a complete incidence matrix. Graph theory, often known as network topology, is concerned with network features that do not change when the geometry of the network changes. The equations created by the graph theory concept (KCL and KVL) are independent equations.

The graph theory concepts make it easier to solve networks with a high number of nodes and branches. The article elaborates on the incidence matrix, properties of the incidence matrix, reduced incidence matrix, and some examples.

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## What is the Incidence Matrix?

In graph theory, while drawing a graph, all the passive elements of a network are represented by lines. The independent voltage and current source are replaced by their internal impedance (independent voltage source-short circuit and independent current source-open circuit).

### Incidence Matrix Definition

The incidence matrix represents a network symbolically. It also makes testing and identifying the independent variables easier. An incidence matrix is a matrix that uniquely represents a graph.

### What are the Dimensions of an Incidence Matrix?

For a graph of n number of nodes and b number of branches, the complete incidence matrix [A]ij will be a rectangular matrix of order n×b.

[A]ij =1; if branch j is associated with node i and oriented away from node i

[A]ij =-1; if branch j is associated with node i and oriented towards from node i

[A]ij  = 0; if branch j is not associated with node i

## How to Find the Incidence Matrix of a Graph?

Consider a network having 6 branches and 4 nodes as shown below. Node d is taken as a reference node. The equivalent graph of the network is also drawn as shown below. The order of the incidence matrix will be 4×6.

## Properties of Incidence Matrix of a Graph

A complete incidence matrix [A]ij a graph has some properties by which one can identify whether the given matrix is a complete incidence matrix or not. These properties are:

• The sum of values of [A]ij of any column is equal to zero.
• For a closed loop system, the determinant of [A]ij is always zero.
• The rank of the complete incidence matrix [A]ij is n-1 where n=number of nodes in the graph.

## Reduced Incidence Matrix of a Graph

The matrix which is obtained from the complete incidence matrix by eliminating one of the rows is known as the reduced incidence matrix. Generally, the reference node is eliminated from the complete incidence matrix to form a reduced incidence matrix.

The reduced incidence matrix is represented as [Ar]ij. Reduced incidence matrix is used to find the number of trees for connected graphs. The number of trees is calculated by taking the determinant of the product of the reduced incidence matrix with its transpose as shown below:

## Incidence Matrix and KCL

Consider a random graph as shown below. The branch currents are i1, i2, i3, i4, i5, and i6 for respective branches. For a graph having n nodes, then, from the incidence matrix we can get n-1 independent KCL equations.

## Incidence Matrix and KVL

Consider the same graph used in the incidence matrix and KCL as shown below.

## Incidence Matrix Example

Example 1: Find the possible number of trees for the graph shown below.

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## FAQs on Incidence Matrix

• An incidence matrix is a matrix that symbolically represents a network. The order of an incidence matrix of n number of nodes and b number of branches is n×b. For each and every graph, the incidence matrix can be obtained.

• A reduced incidence matrix can be obtained by removing one of the rows from the complete incidence matrix. Although, the reference node is removed for simplification purposes.

• For a graph that contains n number of nodes and b number of branches, the dimension of an incidence matrix is n×b. The rank of the incidence matrix is n-1.

• For any graph, the rows of the incidence matrix reflect the number of nodes and the columns of the incidence matrix represents the number of branches. If there are 'n' rows in a particular incidence matrix, this implies there are 'n' nodes in a network.

• From the reduced incidence matrix, the possible number of trees on a graph can be calculated by the determinant of the product of the reduced incidence matrix and the transpose of the reduced incidence matrix.

Number of trees = |[ A_r ][A_r]^T |

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