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A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road.
Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
In formal terms, a directed graph is an ordered pair G = (V, A) where [1]. V is a set whose elements are called vertices, nodes, or points;; A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
The input to the optimal network design problem is a weighted graph G = (V,E), where the weight of each edge (u,v) in the graph represents the cost of building a road from u to v; and a budget B. A feasible network is a subset S of E, such that the sum of w(u,v) for all (u,v) in S is at most B, and there is a path between every two nodes u and ...
Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
A weighted network is a network where the ties among nodes have weights assigned to them. A network is a system whose elements are somehow connected. [1] The elements of a system are represented as nodes (also known as actors or vertices) and the connections among interacting elements are known as ties, edges, arcs, or links.
represents the edge weight between nodes i and j; see Adjacency matrix; and are the sum of the weights of the edges attached to nodes i and j, respectively; m is the sum of all of the edge weights in the graph; N is the total number of nodes in the graph;