To prevent cracks from occurring in thread roots of a socket during the use of a graphite electrode, and facilitate the machining therefor.

In this connecting structure 1 of a graphite electrode for an electric furnace, a socket 51 is provided on an end face of a graphite electrode 53, a nipple 52 is threadably mounted on the socket 51, and another graphite electrode mounted with a socket of the same shape as the socket 51 on an end face thereof is threadably mounted in the nipple 52 and thereby axially connected to the graphite electrode 53. Threads 2 of the socket 51 and/or threads 3 of the nipple 52 are thinned by cutting anti-flank surfaces 2a, 3a thereof, respectively, thereby giving flexibility to the thin-walled threads 2, 3. Force exerted on the flank surfaces 4 of the threads 2 of the socket 51 is relaxed by the flexure of the threads 2, 3, thus avoiding stress concentration on thread roots, and the occurrence and growth of cracks in the thread roots of the socket 51 can be inhibited during the use of the graphite electrode 53.