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.