Eccentric sequence

Eccentric sequence --- последовательность эксцентриситетов.

For a graph [math]\displaystyle{ G }[/math], the eccentric sequence of [math]\displaystyle{ G }[/math] is the sequence of the vertex eccentricities in ascending order. A vertex [math]\displaystyle{ v }[/math] is a mode vertex if the eccentricity of [math]\displaystyle{ v }[/math] appears at least as often in the eccentric sequence as any other eccentricity. The mode of a graph [math]\displaystyle{ G }[/math] is the subgraph induced by its mode vertices.

If [math]\displaystyle{ u }[/math] is a farthest vertex from [math]\displaystyle{ v }[/math], then [math]\displaystyle{ u }[/math] is an eccentric vertex of [math]\displaystyle{ v }[/math], and we say that [math]\displaystyle{ u }[/math] is an eccentric vertex if it is an eccentric vertex of at least one vertex of [math]\displaystyle{ G }[/math]. In a graph [math]\displaystyle{ G }[/math] with a vertex set [math]\displaystyle{ V(G) }[/math], vertices in a set [math]\displaystyle{ X \subseteq V(G) }[/math] are mutually eccentric if for all pairs [math]\displaystyle{ u, v \in X }[/math], [math]\displaystyle{ u }[/math] is eccentric for [math]\displaystyle{ v }[/math] and [math]\displaystyle{ v }[/math] is eccentric for [math]\displaystyle{ u }[/math].

Another way of describing an eccentric vertex is to say that [math]\displaystyle{ u }[/math] is an eccentric vertex of [math]\displaystyle{ v }[/math] if [math]\displaystyle{ d(u,v) = e(v) }[/math]. If each vertex of [math]\displaystyle{ G }[/math] has exactly one eccentric vertex, then [math]\displaystyle{ G }[/math] is unique eccentric point graph. A simple class of unique eccentric point graphs are the paths [math]\displaystyle{ P_{2n} }[/math] on an even number of vertices.