根图的稳定性及其优化

王冰杰,唐晓清

doi:10.13718/j.cnki.xsxb.2017.04.003

西南师范大学学报(自然科学版)

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根图的稳定性及其优化

王冰杰,唐晓清. 根图的稳定性及其优化[J]. 西南师范大学学报(自然科学版), 2017, 42(4). doi: 10.13718/j.cnki.xsxb.2017.04.003

引用本文:

王冰杰,唐晓清. 根图的稳定性及其优化[J]. 西南师范大学学报(自然科学版), 2017, 42(4). doi: 10.13718/j.cnki.xsxb.2017.04.003

WANG Bing-jie,TANG Xiao-qing. On Stability of Rooted Graph and Its Optimality[J]. Journal of Southwest China Normal University(Natural Science Edition), 2017, 42(4). doi: 10.13718/j.cnki.xsxb.2017.04.003

Citation:

WANG Bing-jie,TANG Xiao-qing. On Stability of Rooted Graph and Its Optimality[J]. Journal of Southwest China Normal University(Natural Science Edition), 2017, 42(4). doi: 10.13718/j.cnki.xsxb.2017.04.003

根图的稳定性及其优化

王冰杰,唐晓清

白城师范学院数学与统计学院,吉林白城,137000 ; 上海立信会计金融学院统计与数学学院,上海,201620

On Stability of Rooted Graph and Its Optimality

WANG Bing-jie,TANG Xiao-qing

摘要: 设灾难发生时, 根图G的边以概率p独立幸存, 则含根连通子图的顶点数的期望值EV(G;p)是根图的可靠性的合适指标. 定义了子图的顶点数的平方期望值E2(G;p)后, 则方差D(G;p)=E2(G;p)-[EV(G;p)]2是根图稳定性的合适指标. 推导得到了E2(G;p)的减-缩边公式, 从而得到方差的一个递归计算方法. 进而研究了一些特殊图的方差的计算公式. 最后, 结合期望和方差, 讨论了根图的优化问题.
- 根图 /
- 可靠性 /
- 稳定性 /
- 减-缩边公式 /
- 期望-方差优化
Abstract: When G is a rooted graph where each edge may independently succeed with probability p when catastrophic thing happens, we consider the expected number of vertices in the operational component of G containing the root.Then the expected value of edges number EV(G;p) is a proper index of reliability to rooted graph.Later, we give the definition E2(G;p), which is the expect of vertices number square, then variance D(G;p)=E2(G;p)-[EV(G;p)]2.Especially, we get the deletion-contraction edge formula of E2(G;p).So we obtain a recursive computing variance method.And D(G;p) is a proper stability index to the rooted graph.With this formula, we get some variance computational formulas of specific rooted graphs.Finally, we propose expect-variance optimality of rooted graph.
- rooted graph；reliability；stability；deletion-contraction edge formula；expect-variance optimality /

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根图的稳定性及其优化

王冰杰,唐晓清

白城师范学院数学与统计学院,吉林白城,137000 ; 上海立信会计金融学院统计与数学学院,上海,201620

关键词:

摘要: 设灾难发生时, 根图G的边以概率p独立幸存, 则含根连通子图的顶点数的期望值EV(G;p)是根图的可靠性的合适指标. 定义了子图的顶点数的平方期望值E2(G;p)后, 则方差D(G;p)=E2(G;p)-[EV(G;p)]2是根图稳定性的合适指标. 推导得到了E2(G;p)的减-缩边公式, 从而得到方差的一个递归计算方法. 进而研究了一些特殊图的方差的计算公式. 最后, 结合期望和方差, 讨论了根图的优化问题.

English Abstract

On Stability of Rooted Graph and Its Optimality

白城师范学院数学与统计学院,吉林白城,137000 ; 上海立信会计金融学院统计与数学学院,上海,201620

Keywords:

Abstract: When G is a rooted graph where each edge may independently succeed with probability p when catastrophic thing happens, we consider the expected number of vertices in the operational component of G containing the root.Then the expected value of edges number EV(G;p) is a proper index of reliability to rooted graph.Later, we give the definition E2(G;p), which is the expect of vertices number square, then variance D(G;p)=E2(G;p)-[EV(G;p)]2.Especially, we get the deletion-contraction edge formula of E2(G;p).So we obtain a recursive computing variance method.And D(G;p) is a proper stability index to the rooted graph.With this formula, we get some variance computational formulas of specific rooted graphs.Finally, we propose expect-variance optimality of rooted graph.