Final answer:
In graph theory, a tree's diameter is not equal to twice the radius when the central vertex is not the midpoint of the longest path. The longest path (diameter) exceeds twice the distance from the central vertex to the furthest vertex (radius) in such cases.
Step-by-step explanation:
The question suggests an apparent contradiction between the concepts of the diameter and the radius of a geometrical figure. Specifically, it refers to a tree (in this context, a connected acyclic graph in graph theory), where the diameter of a tree is not simply twice the radius. In a tree graph, the diameter is the longest path between any two vertices, while the radius is the minimum distance from a central vertex (called the root) to the furthest vertex.
For a tree's diameter to be not equal to twice the radius, the central vertex (from which the radius is measured) must not be the midpoint of the longest path in the tree. This occurs in trees where the central vertex is closer to one end of the longest path than to the other. In such cases, doubling the radius does not yield the diameter.
As an example, consider a tree with a central vertex connected to a string of three vertices in a line. The radius would be the distance from the central vertex to the farthest vertex in the line, which is 3 edges. However, the diameter would be the length of the path connecting the two farthest vertices, which is 5 edges (since you must count the edge connecting to the central vertex and then the rest of the path). Here, the diameter is not equal to twice the radius, since 2 x 3 < 5.