High performance computing system, which takes an interconnection network as its infrastructure topology, has been utilized in scientific computing, big data analysis as well as artificial intelligence. With the rapid growth of infrastructure topology (interconnection network) in high performance computing system, the probability of network element malfunction increases dramatically. Generally, the reliability of interconnection network is measured by the traditional (edge) connectivity, which is limited by the minimum degree of the network. In order to overcome the defect, the matroidal connectivity has been newly proposed to extend the edge connectivity. For the regular network $\mathcal{G}$, let ${\mathfrak{B}}_k(\mathcal{G})=\{B_1,B_2,\dots ,B_k\}$ be a fixed edge partition ordered sequence of $\mathcal{G}$ and ${\mathit{\chi }}_{\mathbf{k}}=(x_1,x_2,\dots ,x_k)$ be a sequence of k integers. For any $D\subseteq E(\mathcal{G})$, if $|D\cap B_i|\le x_i$ $(i\in [k])$ and $\mathcal{G}-D$ is disconnected, then D is named matroidal cut set of $\mathcal{G}$. The matroidal connectivity of $\mathcal{G}$, denoted by $\lambda (\mathcal{G},{\mathfrak{B}}_k(\mathcal{G}))$, is the minimum size of matroidal cut set of $\mathcal{G}$. In this work, we characterize the matroidal connectivity of a class of n-dimensional regular networks ${\mathcal{G}}_n$. To be more specific, we show that $\lambda ({\mathcal{G}}_n,{\mathfrak{B}}_n({\mathcal{G}}_n))=(k-l+1)f(l+j)$, where $f(n)$ is a function about n, and $l,k,n$ are positive integers ($l\ge 1$), $j=n-k$. As empirical analysis, we directly determine the matroidal connectivity of some well-known interconnection networks, such as hypercube $Q_n$, star graph $S{T}_n$ and alternating group graph $A{G}_n$.

以互联网络为基础架构拓扑的高性能计算系统已应用于科学计算、大数据分析以及人工智能等领域。随着高性能计算系统中基础设施拓扑（互连网络）的快速增长，网元发生故障的概率急剧增加。通常，互连网络的可靠性是通过传统（边缘）连通性来衡量的，其受到网络最小度的限制。为了克服这一缺陷，新提出了拟阵连通性来扩展边缘连通性。对于常规网络$\mathcal{G}$， 让${\mathfrak{乙}}_k（\mathcal{G}）=\{{乙}_1,{乙}_2,\dots \dots ,{乙}_k\}$是固定边划分有序序列$\mathcal{G}$和${\mathit{\chi }}_{\mathbf{k}}=（X_1,X_2,\dots \dots ,X_k）$是k个整数的序列。对于任何$D\subseteq 乙（\mathcal{G}）$， 如果$|D\cap {乙}_{我}|\le X_{我}$ $（我\epsilon [k]）$和$\mathcal{G}-D$是断开的，则D称为矩阵割集$\mathcal{G}$。矩阵连通性为$\mathcal{G}$，表示为$\lambda （\mathcal{G},{\mathfrak{乙}}_k（\mathcal{G}））$，是矩阵割集的最小尺寸$\mathcal{G}$。在这项工作中，我们描述了一类n维规则网络的拟阵连通性${\mathcal{G}}_n$。更具体地说，我们表明$\lambda （{\mathcal{G}}_n,{\mathfrak{乙}}_n（{\mathcal{G}}_n））=（k-我+1）F（我+j）$， 在哪里$F（n）$是关于n 的函数，并且$我,k,n$是正整数 ($我\ge 1$),$j=n-k$。作为实证分析，我们直接确定了一些著名互连网络的拟阵连通性，例如超立方体${问}_n$, 星图$S{\mathrm{时间}}_n$和交替组图$A{G}_n$。