Search results
Results From The WOW.Com Content Network
An adjacency list representation for a graph associates each vertex in the graph with the collection of its neighbouring vertices or edges. There are many variations of this basic idea, differing in the details of how they implement the association between vertices and collections, in how they implement the collections, in whether they include both vertices and edges or only vertices as first ...
Adjacency lists are generally preferred for the representation of sparse graphs, while an adjacency matrix is preferred if the graph is dense; that is, the number of edges | | is close to the number of vertices squared, | |, or if one must be able to quickly look up if there is an edge connecting two vertices.
In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.
In the remainder of the article it is assumed that the graph is represented using an adjacency matrix. We expect the output of the algorithm to be a distancematrix D {\displaystyle D} . In D {\displaystyle D} , every entry d i , j {\displaystyle d_{i,j}} is the weight of the shortest path in G {\displaystyle G} from node i {\displaystyle i} to ...
Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also used in the clustering coefficient of a graph, which is a measure of the average density of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of ...
where is the (non-square) matrix having elements and is the so-called modularity matrix, which has elements B v w = A v w − k v k w 2 m . {\displaystyle B_{vw}=A_{vw}-{\frac {k_{v}k_{w}}{2m}}.} All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always 0 {\displaystyle 0} .
Efficiency can also be used to determine cost-effective structures in weighted and unweighted networks. [2] Comparing the two measures of efficiency in a network to a random network of the same size to see how economically a network is constructed. Furthermore, global efficiency is easier to use numerically than its counterpart, path length. [5]
Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrix A of a strongly regular graph satisfies two equations. First: = =, which is a restatement of the regularity requirement. This shows that k is an eigenvalue of the adjacency matrix with the all-ones eigenvector.