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The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem: (()) = (). where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and I m is the identity matrix of dimension m.
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.
Adjacency matrix [3] A two-dimensional matrix, in which the rows represent source vertices and columns represent destination vertices. Data on edges and vertices must be stored externally. Only the cost for one edge can be stored between each pair of vertices. Incidence matrix [4]
Matrix structures include the incidence matrix, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix, in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects.
In the case of a graph, the adjacency matrix is a square matrix which indicates whether pairs of vertices are adjacent. Likewise, we can define the adjacency matrix A = ( a i j ) {\displaystyle A=(a_{ij})} for a hypergraph in general where the hyperedges e k ≤ m {\displaystyle e_{k\leq m}} have real weights w e k ∈ R {\displaystyle w_{e_{k ...
In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose, where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example:
The adjacency matrix of a directed graph is a logical matrix, and is unique up to permutation of rows and columns. Another matrix representation for a directed graph is its incidence matrix . See direction for more definitions.
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.