论文标题
复合物，图形，同型，产品和香农容量
Complexes, Graphs, Homotopy, Products and Shannon Capacity
论文作者
论文摘要
有限的抽象简单复合物G定义了Barycentric Refinement Graph Phi（g）=（g，{（a，b），A子集B或B子集A}）和连接图PSI（G）=（G，g，a，b），与b not b notement}相交的）。我们在这里注意到，从复合物到图的函数PHI和PSI在图像（定理1）上都是可逆的，并且G，Phi（g），PSI（G）都有相同的自动形态组，而G的笛卡尔产物对应于Phi（g）的Stanley-Reisner-Reisner产品对应于Stanley-Reisner的产物，并且PSI（G）的Shannon offorciss of Psi（G），已有生产的产品。其次，我们看到如果G是Barycentric的细化，则PHI（G）和PSI（G）是图形同位素（定理2）。第三，如果伽玛是几何实现函数，将其分配给复合物或图形的几何实现，则其集团复合物的几何实现，则gamma（g）和伽马（phi（g））和伽马（PSI（g））都是经典的同型，用于Barycentric精制的Simplecicial Simplecial Complace G（Theoorem 3）。在定理2和3中，必须使用barycentric假设。与复合物的笛卡尔产物具有兼容性，这些配合物在连接图的强图中表现出来：如果两个图A，则A'是同型，B，B，B'是同质的，则是a。 B对A是同质的。 B'（定理4）导致图形同拷贝类的交换环。最后，我们注意到（定理5），对于所有简单络合物g以及产品g = g_1 x g_2 ... x g_k，psi（g）的shannon容量theta（psi（g））等于G。G。G。G。 theta（psi（g））= theta（psi（g））= m使theta与不相交的联合添加和强乘法兼容。
A finite abstract simplicial complex G defines the Barycentric refinement graph phi(G) = (G,{ (a,b), a subset b or b subset a }) and the connection graph psi(G) = (G,{ (a,b), a intersected with b not empty }). We note here that both functors phi and psi from complexes to graphs are invertible on the image (Theorem 1) and that G,phi(G),psi(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of phi(G) and the strong Shannon product of psi(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then phi(G) and psi(G) are graph homotopic (Theorem 2). Third, if gamma is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then gamma(G) and gamma(phi(G)) and gamma(psi(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A,A' are homotopic and B,B' are homotopic, then A . B is homotopic to A' . B' (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G=G_1 x G_2 ... x G_k, the Shannon capacity Theta(psi(G)) of psi(G) is equal to the number m of zero-dimensional sets in G. An explicit Lowasz umbrella in R^m leads to the Lowasz number theta(G) leq m and so Theta(psi(G))=theta(psi(G))=m making Theta compatible with disjoint union addition and strong multiplication.
