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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.
In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the vertices in G is at most d.
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.
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 ...
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 .
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 n {\displaystyle n} points in time O ( n 2 ) {\displaystyle O(n^{2})} (assuming constant-time distance evaluations) but faster algorithms are possible for points in low dimensions.
In graph theory, the McKay–Miller–Širáň graphs are an infinite class of vertex-transitive graphs with diameter two, and with a large number of vertices relative to their diameter and degree. They are named after Brendan McKay, Mirka Miller, and Jozef Širáň, who first constructed them using voltage graphs in 1998. [1]
In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given ...