Document Type : Original Article
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, P. O. Box 1159, Mashhad 91775, Iran.
A graph is called $3$-self-centered if all of its vertices have eccentricity $3.$ In this paper,
we study some properties of the $3$-self-centered graphs, and
investigate the $3$-self-centered graphs with girth six or seven. We determine
the vertex connectivity and regularity
of such graphs. Moreover, we show that if $G$ is a $3$-self-centered graph with girth seven, then
$G$ is regular.