Fundamental groups of clique complexes of random graphs
Fundamental groups of clique complexes of random graphs
复制标题
随机图派系复合体的基本群
DOI：
10.1112/tlms/tlv001
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发表时间：
2015
影响因子：
0.8
通讯作者：
Costa A
中科院分区：
文献类型：
--
作者：
Costa A
We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs. We establish thresholds for a number of properties of fundamental groups of these complexes. In particular, if, then we show that \[\begin {matrix}{{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma ))=1 &\mbox {if}\ \alpha \lt -\tfrac {1}{2}, \\ {{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma )) =2 & \mbox {if}\ -\tfrac {1}{2} \lt \alpha \lt -\tfrac {11}{30}, \\ {{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma )) =\infty & \mbox {if}\ -\tfrac {11}{30} \lt \alpha \lt -\tfrac {1}{3}, \end {matrix}\] asymptotically almost surely (a.a.s.), whereanddenote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental groupis trivial for. We prove that forthe fundamental grouphas 2-torsion but has no-torsion for any given prime. We also prove that aspherical subcomplexes of the random clique complexsatisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.
We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs. We establish thresholds for a number of properties of fundamental groups of these complexes. In particular, if, then we show that \[\begin {matrix}{{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma ))=1 &\mbox {if}\ \alpha \lt -\tfrac {1}{2}, \\ {{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma )) =2 & \mbox {if}\ -\tfrac {1}{2} \lt \alpha \lt -\tfrac {11}{30}, \\ {{\rm gdim}}(\pi _1(X_\Gamma ))={{\rm cd}}(\pi _1(X_\Gamma )) =\infty & \mbox {if}\ -\tfrac {11}{30} \lt \alpha \lt -\tfrac {1}{3}, \end {matrix}\] asymptotically almost surely (a.a.s.), whereanddenote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental groupis trivial for. We prove that forthe fundamental grouphas 2-torsion but has no-torsion for any given prime. We also prove that aspherical subcomplexes of the random clique complexsatisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.
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