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:
“Small, black, as flies hanging in heat, the Boys,
Until the distance throws them forth, their hum
Bulges to thunder held by calf and thigh.”
—Thom Gunn (b. 1929)
“Are we not madder than those first inhabitants of the plain of Sennar? We know that the distance separating the earth from the sky is infinite, and yet we do not stop building our tower.”
—Denis Diderot (1713–1784)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)