incidensmatrise

Incidensmatrise is a mathematical concept used in graph theory to represent the relationship between the vertices and edges of a graph. It is a matrix where the rows correspond to the vertices of the graph and the columns correspond to the edges. The entries in the matrix indicate whether a particular edge is incident to a particular vertex.

For a simple graph with $n$ vertices and $m$ edges, the incidence matrix $B$ is an $n \times m$ matrix. An entry $B_{ij}$ is typically set to 1 if vertex $i$ is an endpoint of edge $j$, and 0 otherwise. For directed graphs, the convention may differ; for instance, a $+1$ might indicate an outgoing edge from the vertex, and a $-1$ might indicate an incoming edge. In some contexts, a $1$ is used for an incident vertex and a $-1$ for the vertex where the edge terminates.

The incidence matrix provides a clear and structured way to study graph properties. For example, the sum of each column in the incidence matrix for an undirected graph is always 2, as each edge connects exactly two vertices. The sum of each row indicates the degree of the corresponding vertex. This matrix is particularly useful in network flow problems, electrical circuit analysis, and other areas where the connections between elements are crucial. It serves as a fundamental tool for translating graph structures into a format suitable for linear algebra operations.