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For a signed graph $G$ and non-negative integer $d$, it was shown by DeVos et al. that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where $\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We define a class of  particular directed circuits in $G$, namely the fundamental directed circuits, and show that all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly partitioned into $2^{\epsilon(\Gamma)}$ classes specified by the elements of order 2 in $\Gamma$, each class of which consists of the same number of flows depending only on the order of  $\Gamma$. Using an extension of  Whitney's broken circuit theorem of Dohmen and Trinks, we give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for $d=0$ in terms of broken bonds. Finally,  we show that the sets of edges  in a signed graph that contain no broken bond form a  homogeneous  simplicial complex.

For the multiprocessor systems modeled by interconnection networks, one of the important properties is the characterization of fault tolerability. Connectivity, as an important parameter to evaluate fault tolerability, has witnessed research achievements. To make the evaluation more practical, conditional connectivity has been promisingly proposed. As one kind of conditional connectivity, $h$-restricted connectivity of a connected graph $G$, denoted by $\kappa ^h (G)$, is defined as the cardinality of the minimum vertex cut set $F$ such that $\delta (G-F)\geq h$. In this paper, we establish a universally $h$-restricted connectivity for a class of hypercube-based compound networks, in which the well-known networks, such as hierarchical cubic network $HCN(n, n)$ and its generalization complete cubic network $CCN(n)$, are involved.

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