On eccentric vertices in graphs

Abstract

The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex furthest from v. The minimum eccentricity among the vertices of G is the radius rad G of G, and the maximum eccentricity is its diameter diam G. A vertex u of G is called an eccentric vertex of v if d(u, v) = e(v). The radial number m(v) of v is the minimum eccentricity among the eccentric vertices of v, while the diametrical number dn(v) of v is the maximum eccentricity among the eccentric vertices of v. The radial number m(G) of G is the minimum radial number among the vertices of G and the diametrical number dn(G) of G is the minimum diametrical number among the vertices of G. Several results concerning eccentric vertices are presented. It is shown that for positive integers a and b with a ≤ b ≤ 2a there exists a connected graph G having m(G) = a and dn(G) = b. Also, if a, b, and c are positive integers with a ≤ b ≤ c ≤ 2a, then there exists a connected graph G with rad G = a, m(G) = b, and diam G = c. © 1996 John Wiley & Sons, Inc.