Incidence matrix vs Adjacent Matrix

In summary, the difference between adjacent matrices and incidence matrices lies in their definition and use in different types of graphs. While adjacent matrices are used for undirected graphs and are symmetric, incidence matrices are used for directed graphs and can be either symmetric or asymmetric. The definition of incidence matrices can also vary, with some authors considering it as the transpose of the traditional definition. Kirchhoff was the first to introduce the concept of incidence matrices.
  • #1
John Creighto
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What is the difference between an http://en.wikipedia.org/wiki/Incidence_matrix" . They sound the same to me but this paper says they are different:
http://eprints.pascal-network.org/archive/00005332/01/barber_Newton.pdf

edit: My guess is that adjacent matrices refer to non directed graphs so the matrix will be symmetric while an incidence matrix also incidence directed graphs so need not be symmetric.
 
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  • #2
The deffinition at mathworld clarified it for me:

The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this, with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.​
http://mathworld.wolfram.com/IncidenceMatrix.html
 

What is the difference between an incidence matrix and an adjacent matrix?

An incidence matrix is a mathematical representation of a graph where the rows represent the vertices and the columns represent the edges. The values in the matrix indicate whether a particular vertex is connected to a particular edge or not. On the other hand, an adjacent matrix is a mathematical representation of a graph where the rows and columns represent the vertices, and the values in the matrix indicate whether two vertices are connected by an edge or not.

Which matrix is more suitable for analyzing the properties of a graph?

Both incidence matrix and adjacent matrix can be used to analyze graph properties, but each has its advantages. Incidence matrix is more suitable for analyzing properties related to edges, such as the number of edges incident to a vertex or the degree of a vertex. On the other hand, adjacent matrix is more suitable for analyzing properties related to vertex connectivity, such as finding the shortest path between two vertices or determining whether a graph is connected or not.

How do I convert an adjacent matrix to an incidence matrix?

To convert an adjacent matrix to an incidence matrix, you can follow these steps: 1) Create a matrix with the same dimensions as the adjacent matrix, but with all values set to 0. 2) For each non-zero value in the adjacent matrix, find the corresponding row and column indices. 3) Set the value in the same row and column of the incidence matrix to 1. Repeat for all non-zero values in the adjacent matrix.

Can an incidence matrix or an adjacent matrix be used to represent a directed graph?

Yes, both incidence matrix and adjacent matrix can be used to represent directed graphs. In an incidence matrix, the values in the matrix can indicate the direction of the edge (e.g. 1 for outgoing edge, -1 for incoming edge). In an adjacent matrix, the values can also indicate the direction of the edge (e.g. 1 for an edge going from row vertex to column vertex, 0 for no connection, -1 for an edge going from column vertex to row vertex).

What are the applications of incidence matrix and adjacent matrix in real-world problems?

The incidence matrix and adjacent matrix are widely used in various fields such as computer science, engineering, and social sciences. Some common applications include network analysis, transportation and logistics planning, circuit design, and social network analysis. They are also used in data structures and algorithms for efficient graph representation and manipulation.

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