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. There are variations; see below.
Graph theory Incidence matrix is a common graph representation in graph theory. It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs.
Undirected and directed graphs
In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented. The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a n × m {\displaystyle n\times m} matrix B, where n and m are the numbers of vertices and edges respectively, such that
B i j = { 1 if vertex v i is incident with edge e j , 0 otherwise. {\displaystyle B_{ij}={\begin{cases}1&{\text{if vertex }}v_{i}{\text{ is incident with edge }}e_{j},\\0&{\text{otherwise.}}\end{cases}}}
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, e 1 , e 2 , e 3 , e 4 {\displaystyle e_{1},e_{2},e_{3},e_{4}} ):
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end. The incidence matrix of a directed graph is a n × m {\displaystyle n\times m} matrix B where n and m are the number of vertices and edges respectively, such that
B i j = { − 1 if edge e j leaves vertex v i , − 1 if edge e j enters vertex v i , − 0 otherwise. {\displaystyle B_{ij}={\begin{cases}{-1}&{\text{if edge }}e_{j}{\text{ leaves vertex }}v_{i},\\{\phantom {-}}1&{\text{if edge }}e_{j}{\text{ enters vertex }}v_{i},\\{\phantom {-}}0&{\text{otherwise.}}\end{cases}}}
(Many authors use the opposite sign convention.) The oriented incidence matrix of an undirected graph is the incidence matrix, in the sense of directed graphs, of any orientation of the graph. That is, in the column of edge e, there is one 1 in the row corresponding to one vertex of e and one −1 in the row corresponding to the other vertex of e, and all other rows have 0. 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 unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:
A ( L ( G ) ) = B ( G ) T B ( G ) − 2 I m . {\displaystyle A(L(G))=B(G)^{\textsf {T}}B(G)-2I_{m}.}
where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and Im is the identity matrix of dimension m. The discrete Laplacian (or Kirchhoff matrix) is obtained from the oriented incidence matrix B(G) by the formula
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