Central vertex

Definition

Suppose $G$ is a graph of finite diameter, i.e., it is a connected undirected graph that has a finite radius. (Note that any finite connected undirected graph must have finite radius, but infinite graphs may sometimes have finite radius). A vertex $v$ of $G$ is termed a central vertex if the eccentricity of $v$ (i.e., the maximum of distances between $v$ and vertices of $G$) equals the radius of $G$. Note that the radius is defined as the minimum of the eccentricities of all vertices.

Facts

• Tree has either one central vertex or two adjacent central vertices
• Connected graph may have precisely two non-adjacent central vertices

Related properties

• Peripheral vertex is in some sense the opposite: it is a vertex whose eccentriciti equals the diameter, which is the maximum of all eccentricities.