What is incidence matrix?

The incidence matrix is a mathematical representation of a directed or undirected graph, in which the rows represent the edges and the columns represent the vertices. Thus, each element of the matrix indicates whether a particular vertex is connected to a particular edge or not.

In the case of a directed graph, the incidence matrix will be a rectangular matrix with rows representing the edges and columns representing the vertices. The entry in edge-row i and vertex-column j of the matrix will be 1 if the edge i is incident on the vertex j (i.e., the edge goes from j to some other vertex), -1 if the edge i is incident on vertex j (i.e., the edge comes from j), and 0 otherwise.

In the case of an undirected graph, the incidence matrix will also be a rectangular matrix, but with rows representing the edges and columns representing the vertices. The entry in edge-row i and vertex-column j of the matrix will be 1 if the edge i is incident on the vertex j, and 0 otherwise.

The incidence matrix is a useful tool in graph theory, and can be used to solve a variety of problems related to graphs, such as determining the degree of a vertex, finding paths and cycles in a graph, and determining the connectivity of a graph.