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Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph. [2] Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti ...
The nullity of a graph in the mathematical subject of graph theory can mean either of two unrelated numbers. If the graph has n vertices and m edges, then: In the matrix theory of graphs, the nullity of the graph is the nullity of the adjacency matrix A of the graph. The nullity of A is given by n − r where r is the rank of the adjacency matrix.
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M ; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f ) and the nullity of f (the dimension of the kernel of f ).
As a consequence, a rank-k matrix can be written as the sum of k rank-1 matrices, but not fewer. The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix. (This is the rank–nullity theorem.) If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are equal.
The circuit rank of a hypergraph can be derived by its Levi graph, with the same circuit rank but reduced to a simple graph. = + (+) where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components.
For a planar graph, the nullity is equal to the number of meshes in the graph. [34] The rank, R of a graph is defined by: = Rank plays the same role in nodal analysis as nullity plays in mesh analysis. That is, it gives the number of node voltage equations required. Rank and nullity are dual concepts and are related by: [35]
A set is a basis if its rank equals both its cardinality and the rank of the matroid. [3] A set is closed if it is maximal for its rank, in the sense that there does not exist another element that can be added to it while maintaining the same rank. The difference | | is called the nullity of the subset .
Hence the examples above represent zero matrices over any ring. The zero matrix also represents the linear transformation which sends all the vectors to the zero vector. [5] It is idempotent, meaning that when it is multiplied by itself, the result is itself. The zero matrix is the only matrix whose rank is 0.