Search results
Results from the WOW.Com Content Network
In graph theory, the diameter of a connected undirected graph is the farthest distance between any two of its vertices. That is, it is the diameter of a set for the set of vertices of the graph, and for the shortest-path distance in the graph. Diameter may be considered either for weighted or for unweighted graphs.
A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric. The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected. The eccentricity ϵ(v) of a vertex v is the greatest distance between v and any other vertex; in ...
The size of G is bounded above by the Moore bound; for 1 < k and 2 < d, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.
The diameter is always attained by two points of the convex hull of the input. A trivial brute-force search can be used to find the diameter of points in time () (assuming constant-time distance evaluations) but faster algorithms are possible for points in low dimensions.
In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter.The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer.
The metric dimension of an n-vertex graph is n − 2 if and only if the graph is a complete bipartite graph K s, t, a split graph + ¯ (,), or + (,). Relations between the order, the metric dimension and the diameter
For the metric space of shortest-path distances in a graph, a minimum-diameter spanning tree can also be a spanning tree of the graph, a tree whose edges all belong to the graph. However, this may require it to have more than two non-leaf vertices. In this case, the problem is equivalent to finding an absolute 1-center of the graph. This is a ...
In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is d and its diameter is k , its girth must equal 2 k + 1 .