The distance dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest path between them. The subscript G is usually dropped when there is no danger of confusion. When u and v are identical, their distance is 0. When u and v are unreachable from each other, their distance is defined to be infinity ∞.
The eccentricity εG(v) of a vertex v in a graph G is the maximum distance from v to any other vertex. The diameter diam(G) of a graph G is the maximum eccentricity over all vertices in a graph; and the radius rad(G), the minimum. When there are two components in G, diam(G) and rad(G) defined to be infinity ∞. Trivially, diam(G) ≤ 2 rad(G). Vertices with maximum eccentricity are called peripheral vertices. Vertices of minimum eccentricity form the center. A tree has at most two center vertices.
The Wiener index of a vertex v in a graph G, denoted by WG(v) is the sum of distances between v and all others. The Wiener index of a graph G, denoted by W(G), is the sum of distances over all pairs of vertices. An undirected graph's Wiener polynomial is defined to be Σ qd(u,v) over all unordered pairs of vertices u and v. Wiener index and Wiener polynomial are of particular interest to mathematical chemists.
The k-th power Gk of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G with distance at most k. A second power of a graph is also called a square.
A k-spanner is a spanning subgraph, S, in which every two vertices are at most k times as far apart on S than on G. The number k is the dilation. k-spanner is used for studying geometric network optimization.
Read more about this topic: Glossary Of Graph Theory
Famous quotes containing the word distance:
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—Ralph Waldo Emerson (1803–1882)
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—Franz Grillparzer (1791–1872)
“Like a man traveling in foggy weather, those at some distance before him on the road he sees wrapped up in the fog, as well as those behind him, and also the people in the fields on each side, but near him all appears clear, though in truth he is as much in the fog as any of them.”
—Benjamin Franklin (1706–1790)