On detour self-centered graph

Abstract EN

A connected graph G is detour self-centered if and only if G its own detour center.

Chartrand, et al [2] conjectured that if G is a detour self-centered graph of order p , then the detour eccentricity of each vertex v of G is p -1.

In this paper, we prove this conjecture for graphs G of order p , 3 ≤ p ≤ 8.It is also proved that if G is of order p , 3 ≤ p ≤ 8, then G is detour self-centered if and only if G is Hamiltonian.

A detour selfcentered graph of order 9 which is neither Hamiltonian nor hyperhamiltonian is given in this paper.