![]() ![]() ![]() The main alternative data structure, also in use for this application, is the adjacency list. The adjacency matrix may be used as a data structure for the representation of graphs in computer programs for manipulating graphs. The adjacency matrix can be used to determine whether or not the graph is connected. We divide by 6 to compensate for the overcounting of each triangle (3! = 6 times). A great example of how this is useful is in counting the number of triangles in an undirected graph G, which is exactly the trace of A 3 divided by 6. If n is the smallest nonnegative integer, such that for some i, j, the element ( i, j) of A n is positive, then n is the distance between vertex i and vertex j. If A is the adjacency matrix of the directed or undirected graph G, then the matrix A n (i.e., the matrix product of n copies of A) has an interesting interpretation: the element ( i, j) gives the number of (directed or undirected) walks of length n from vertex i to vertex j. Such linear operators are said to be isospectral. However, two graphs may possess the same set of eigenvalues but not be isomorphic. These can therefore serve as isomorphism invariants of graphs. In particular, A 1 and A 2 are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix, which contains information about the degree of each vertex.įor a simple graph with vertex set U = The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. all of its edges are bidirectional), the adjacency matrix is symmetric. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. Square matrix used to represent a graph or network ![]()
0 Comments
Leave a Reply. |