Matrix of ones

For an n × n matrix of ones J, the following properties hold: When J is considered as a matrix over the real numbers, the following additional properties hold: The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory.

For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA.

[7] As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.

Central groupoids are algebraic structures that obey the identity

Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.