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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
Rank–nullity theorem. 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 ...
Thus A T x = 0 if and only if x is orthogonal (perpendicular) to each of the column vectors of A. It follows that the left null space (the null space of A T) is the orthogonal complement to the column space of A. For a matrix A, the column space, row space, null space, and left null space are sometimes referred to as the four fundamental subspaces.
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 number of the graph. The sum of the rank and the nullity is the number of edges.
An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T). The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
In mathematics, a Sylvester domain, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix, and B is an n by s matrix over R, then ρ(AB) ≥ ρ(A) + ρ(B) – n. where ρ is the inner rank of a matrix.
Nullity (linear algebra), the dimension of the kernel of a mathematical operator or null space of a matrix; Nullity (graph theory), the nullity of the adjacency matrix of a graph; Nullity, the difference between the size and rank of a subset in a matroid; Nullity, a concept in transreal arithmetic denoted by Φ, or similarly in wheel theory ...
Range minimum query reduced to the lowest common ancestor problem. Given an array A[1 … n] of n objects taken from a totally ordered set, such as integers, the range minimum query RMQ A (l,r) =arg min A[k] (with 1 ≤ l ≤ k ≤ r ≤ n) returns the position of the minimal element in the specified sub-array A[l … r].