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For sparse graphs, that is graphs with few edges, an adjacency list is often the preferred representation because it uses less space. An alternative matrix representation for a graph is the incidence matrix.
The relationship between a graph and its adjacency matrix is studied in spectral graph theory.
The adjacency matrix for the example graph
is
The adjacency matrix of an undirected graph is symmetric, and therefore has a complete set of eigenvalues and orthogonal eigenvector basis. The eigenvalues of a graph are the Spectrum of the Graph.
Suppose two directed or undirected graphs G1 and G2 with adjacency matrices A1 and A2 are given. G1 and G2 are isomorphic if and only if there exists a permutation matrix P such that
In particular, A1 and A2 are similar and have therefore the same minimal polynomialThe minimal polynomial of an n by n matrix A over a field F is the monic polynomial p ''x over F of least degree such that p ''A 0. Any other non-zero polynomial f with p ''A 0 is a multiple of p''. The following three statements are equivalent: #λ, characteristic polynomialIn linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial . This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace. Motivation Given a squa, eigenvalues, determinantIn linear algebra, the determinant is a function that associates a scalar det A to every square matrix A''. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants and traceIn linear algebra, the trace of an n by n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A i. tr A A + A +. where A represents the i ''j 'th element of A. The use of t. These can therefore serve as isomorphism invariants of graphs. The multiplication with the permutation matrix can be visualized as a relabeling of the vertices.
If A is the adjacency matrix of the directed or undirected graph G, then the matrix An (i.e. the matrix productThis article gives an overview of the various ways to multiply matrices. The Einstein notation is used throughout. Ordinary matrix product By far the most important way to multiply matrices is the usual matrix multiplication. It is defined between two mat of n copies of A) has an interesting interpretation: the entry in row i and column j gives the number of (directed or undirected) paths of length n from vertex i to vertex j.
The matrix I−A (where I denotes the n-by-n identity matrixIn linear algebra, the identity matrix of size n is the n by n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I or simply by I if the size is immaterial or can be trivially determined by the context. The important prope) is invertible if and only if there are no directed cycles in the graph G. In this case, the inverse (I−A)−1 has the following interpretation: the entry in row i and column j gives the number of directed paths from vertex i to vertex j (which is always finite if there are no directed cycles). This can be understood using the geometric series for matrices:
corresponding to the fact that the number of paths from i to j equals the number of paths of length 0 plus the number of paths of length 1 plus the number of paths of length 2 etc. NOTE :: The main diagonal of every Adjacency matrix has all of its entries as zeroes.(By R.K.uppal)
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