The Classes Of Uniformly Opitimally Graphs In The Complete Tripatite And The Complete 5-partite Graphs

This thesis investigates the synthesis problem of uniformly optimally reliable network whose edges never fail but whose vertices break down independently of eachother (UOR: Uniformly Optimally Reliable). Namely, let Ω(n,m) be the class of all graphs with n nodes and m edges, we attempt to find the most reliable graphs G(n, m),in the class of the graphs Ω(n,m), for all choices of p. The latest research shows thatsome complete tripartite graphs are uniformly optimally reliable in its class. This thesis proves that a new complete tripartite graph is uniformly best in its class. Furthermore, there are two kinds of uniformly optimally reliable networks in complete 5-partite graphs. The conclusions are as follows:(1) The thesis proves there also exists a new class of uniformly optimally reliable network. More precisely, it proves that for any positive integer b(b ≥ 2), the complete tripartite graph K_b,b+2,b+2 is uniformly optimally in its class Ω(3b +4,3b2+8b + 4),while the complete tripartite graphs K_b,b+2,b+j(j > 3) are not the uniformly optimally inthere class Ω(36 + 2 + j,3b2 + 2(2 + j)b + 2j).(2) The thesis proves there also exist two new classes of uniformly optimally reliable networks. Precisely, it proves that for any positive integer b( b ≥2), the complete 5-partite graph K_{b,b+1,b+1,b+1,b+2} is uniformly optimally in its classΩ(5b + 5,10b² +20b + 9), the complete 5-partite graph K_{b,b+1,b+2,b+2} is uniformly optimally in its class Ω(5b + 6,10b2 +24b + 13), while the complete 5-partite graphs K_{b,b+1 b+1,b+1,b+j} ( J > 2 ) are not the uniformly optimally in there class Ω(5b+3+j,10b² +(12+4j)b+3+3j)), the complete 5-partite graphs K_{b,b+1,b+2,b+j}(j>2)are not the uniformly optimally in there class Q(5b + 4 + j,l0b2 + (16 + 4J)b + 6 + Aj).