对称图的均匀分解的推广及应用
Extensions and Applications of Equitable Decompositions for Graphs with Symmetries
摘要：
我们扩展了公平分解的理论，其中，如果一个图有一个特定类型的对称性，即一个统一的或基本的自同构$ϕ$，它是可能的使用$ϕ$分解矩阵$M$适当地与图形。结果是一些严格较小的矩阵，其集合特征值与原始矩阵$M$的特征值相同。我们在这里表明，一个大类的自同构，我们称之为\emph{separable}，可以实现为一个序列的基本自同构，使我们能够公平地分解$M$在任何这样的自同构。我们还表明，不仅可以矩阵$M$被分解，但$M$的特征向量也可以公平地分解。此外，我们证明了在温和的条件下，如果矩阵$M$是公平分解的结果因子矩阵，这是相关的公平划分的因子矩阵，将具有相同的谱半径作为原始矩阵$M$。最后，我们描述了一个公平的分解如何影响的Gershgorin区域$Γ(M)$的矩阵$M$，它可以用来本地化的特征值$M$。我们证明了$M$的公平分解的Gershgorin区域包含在原始矩阵的Gershgorin区域$Γ(M)$中。我们在一个真实的网络上证明，通过一系列的公平分解，可以显着减少矩阵的大小' Gershgorin区域。
英文摘要：
We extend the theory of equitable decompositions, in which, if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism $ϕ$, it is possible to use $ϕ$ to decompose a matrix $M$ appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. We show here that a large class of automorphisms, which we refer to as \emph{separable}, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose $M$ over any such automorphism. We also show that not only can a matrix $M$ be decomposed but that the eigenvectors of $M$ can also be equitably decomposed. Additionally, we prove under mild conditions that if a matrix $M$ is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix $M$. Last, we describe how an equitable decomposition effects the Gershgorin region $Γ(M)$ of a matrix $M$, which can be used to localize the eigenvalues of $M$. We show that the Gershgorin region of an equitable decomposition of $M$ is contained in the Gershgorin region $Γ(M)$ of the original matrix. We demonstrate on a real-world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix' Gershgorin region.
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