马天龙,梁晓东

• 1:新疆大学数学与系统科学学院

摘要(Abstract)：

以(n,m)表示具有n个顶点m条边的图的集合.假设图G的边可靠,而顶点可靠的独立概率为p,若对于所有1 p∈(0,1),图G均为(n,m)中的最可靠图,则称G为一致最优图.本文证明了完全k-部图K(b,(b+2)k 1)在其图类中是一致最优的,而当i>3时,完全k-部图K(b,(b+2)k 2,b+i)在其图类中不是一致最优的.

关键词(KeyWords)： 网络可靠性;完全多部图;一致最优图

Abstract:

Keywords:

基金项目(Foundation): 国家自然科学基金(11171283);; 新疆大学博士启动基金(090106)

作者(Author): 马天龙,梁晓东

参考文献(References)：

• [1]Colbourn C.The Combinatorics of Network Reliability[M].New York:Oxford University Press,1987.
• [2]Shier D R.Network reliability and algebraic structures[M].Oxford:University Press,1991.
• [3]Amin A F,Slegrist K T,Slater P J.On the nonexistence of uniformly optimal graph for pair-connected reliability[J].Networks,1991,21:359-368.
• [4]Boesch F,Li X,Suffel C.On the existence of uniformly optimally reliable networks[J].Networks,1991,21:181-194.
• [5]Gross D,Saccoman J.Uniformly optimally reliable graphs[J].Networks,1998,31:217-225.
• [6]Myrvold W,Cheung K,Page L,et al.Uniformly-most reliable networks do not always exist[J].Networks,1991,21:417-419.
• [7]Petingi L,Saccoman J,Schoppmann L.Uniformly least reliable graphs[J].Networks,1996,27:125-131.
• [8]Boesch F,Satyanarayana A,Suffel C.On residual connectedness network reliability,in:DIMACS Series in Discrete Math-ematics and Theoretical Computer Science[J].American Mathematical Society,1991,5:51-59.
• [9]Goldschmidt O,Jaillet P,Lasota R.On reliablity of graphs with node failures[J].Networks,1994,24:251-259.
• [10]Liu S,Chen K,Liu X.Network reliablity with node failures[J].Networks,2000,35:109-117.
• [11]Yu S,Shao F,Meng H.Uniformly optimal graphs in some classes of graphs with node failures[J].Discrete Mathematics,2010,310:159-166.
• [12]Harary F.The maximum connectivity of a graph[J].Proc Nat Acad Sci U S A,1962,48:1142-1146.
• [13]Boesch F,Fclzer A.On the invulnerability of the regular complete k-partite graphs[J].SIAM J Appl Math,1971,20:176-182.
• [14]Stivaros C.On the residual node connectedness network reliability model[D].Ph.D.Dissertation,Dept of Electrical Engi-neering and Computer Science,Stevens Institute of Technology,1990.
• [15]Liang X D,Meng J X,Zhang Z.Uniformly optimal complete multi-partite graphs[J].Journal of Xinjiang University,2012,29:1-8.