Incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation.
If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each mapping from X to Y.
The entry in row x and column y is 1 if the vertex x is part of (called incident in this context) the mapping that corresponds to y, and 0 if it is not.
It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs.
matrix B, where n and m are the numbers of vertices and edges respectively, such that For example, the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges,
If we look at the incidence matrix, we see that the sum of each column is equal to 2.
matrix B where n and m are the number of vertices and edges respectively, such that (Many authors use the opposite sign convention.)
The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.
The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph.
The definitions of incidence matrix apply to graphs with loops and multiple edges.
The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries.
By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.
Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design.
The incidence matrix is an important tool in the theory of block designs.
For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.
