In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27--37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.

翻译：本文考虑直径为2的无三角形图。若直径2的无三角形图$G$不同构于一个星图，则$G$的半径也为2，此类图亦称为2-自中心图。Shekarriz等人[《2-自中心图的一个刻画》，Discuss. Math. Graph Theory 38 (2018), 27--37]给出了2-自中心图的一个刻画，但其刻画存在细微缺陷。因此，本文修正该刻画并证明了此类图的精确刻画。此外，利用我们的刻画，我们证明了关于直径2的无三角形图色数的一些结果。