Abstract: The neighbor expanded sum distinguishing total coloring of join of two cycles was constructed, then the neighbor expanded sum distinguishing total colorings and neighbor expanded sum distinguishing total coloring indexes of joins of one cycle and one path, and two paths were obtained, respectively by deleting edges. The results illustrate that the neighbor expanded sum distinguishing total coloring index of several classes of graphs are all equal to 2; NESDTC conjecture is true for the joins of two cycles, one cycle and one path, and two paths.

Key words: join of graphs, NESDTC conjecture, neighbor expanded sum distinguishing total coloring, neighbor expanded sum distinguishing total coloring index