Distance-balanced closure of some graphs

In this paper we prove that any distance-balanced graph $G$ with $\mathrm{\Delta }(G)\ge \mathrm{\mid }V(G)\mathrm{\mid }-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $\mathrm{\Delta }(T)\ge \mathrm{\mid }V(T)\mathrm{\mid }-3$.