The Star-Structure Connectivity and Star-Substructure Connectivity of Hypercubes and Folded Hypercubes

As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $\kappa (G; T)$ (resp. $T$-substructure connectivity $\kappa ^{s}(G; T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. showed $\kappa (Q_{n};K_{1,1})=\kappa ^{s}(Q_{n};K_{1,1})=n-1$ and $\kappa (Q_{n};K_{1,r})=\kappa ^{s}(Q_{n};K_{1,r})=\lceil \frac{n}{2}\rceil $ for $2\leq r\leq 3$ and $n\geq 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107). Sabir et al. obtained that $\kappa (Q_{n};K_{1,4})=\kappa ^{s}(Q_{n};K_{1,4})= \lceil \frac{n}{2}\rceil $ for $n\geq 6$ and for $n$-dimensional folded hypercube $FQ_{n}$, $\kappa (FQ_{n};K_{1,1})=\kappa ^{s}(FQ_{n};K_{1,1})=n$, $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ with $2\leq r\leq 3$ and $n\geq 7$ (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44–55). They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $r\geq 2$, $\kappa (Q_{n};K_{1,r})$ $=\kappa ^{s}(Q_{n};K_{1,r})$ $=\lceil \frac{n}{2}\rceil $ and $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.

Minimum Linear Arrangement of the Cartesian Product of Optimal Order Graph and Path

The minimum linear arrangement of an arbitrary graph is the embedding of the vertices of the graph onto the line topology in such a way that the sum of the distances between adjacent vertices in the graph is minimized. This minimization can be attained by finding an optimal ordering of the vertex set of the graph and labeling the vertices of the line in that order. In this paper, we compute the minimum linear arrangement of the Cartesian product of certain sequentially optimal order graphs which include interconnection networks such as hypercube, folded hypercube, complete Josephus cube and locally twisted cube with path and the edge faulty counterpart of sequentially optimal order graphs.

Fault-free Hamiltonian cycle including given edges in folded hypercubes with faulty edges

The folded hypercube is an important interconnection network for multiprocessor systems. Let [Formula: see text] with [Formula: see text] denote an [Formula: see text]-dimensional folded hypercube. For a given fault-free edge set [Formula: see text] with [Formula: see text] and a faulty edge set [Formula: see text] with [Formula: see text], in this paper we prove that [Formula: see text] contains a fault-free Hamiltonian cycle including each edge of [Formula: see text] if and only if the subgraph induced by [Formula: see text] is linear forest. Furthermore, we give the definitions of the distance among three vertex-disjoint edges and the distance between a vertex and a vertex set. For three vertex-disjoint edges [Formula: see text], the distance among them is denoted by [Formula: see text]. For a vertex [Formula: see text] and a vertex set [Formula: see text], the distance between [Formula: see text] and [Formula: see text] is denoted by [Formula: see text].

The Connectivity of Exchanged Folded Hypercube

Based on exchanged hypercube, a new interconnection network, exchanged folded hypercube, was proposed recently. It has better properties than other variations of hypercube in some areas. In this work we will show that its connectivity and edge-connectivity are equal to its minimum degree.

Automorphisms and isomorphisms of enhanced hypercubes

Let Zn2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1,..., en} be the standard basis of Zn2 and ?k = ek +...+ en for some 1 ? k ? n-1. Denote by ?n,k the Cayley graph over Zn2 with generating set Sk = {e1,..., en,?k}, that is, ?n,k = Cay(Zn2,Sk). In this paper, we characterize the automorphism group of ?n,k for 1 ? k ? n-1 and determine all Cayley graphs over Zn2 isomorphic to ?n,k. Furthermore, we prove that for any Cayley graph ? = Cay(Zn2,T), if ? and ?n,k share the same spectrum, then ? ? ?n,k. Note that ?n,1 is known as the so called n-dimensional folded hypercube FQn, and ?n,k is known as the n-dimensional enhanced hypercube Qn,k.

Conditional Matching Preclusion for Folded Hypercubes

Let G be a graph with an even number of vertices. The matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion results in a graph with no isolated vertices and without a perfect matching. Matching preclusion number was introduced for measuring the robustness of a network when there is a link failure. In this paper, we focus on conditional matching preclusion for folded hypercube FQn, an important variant of hypercube. We show that conditional matching preclusion number of FQn is 2n and all optimal conditional matching preclusion sets are trivial for n ⩾ 5.

Spectra analysis and network coherence for weighted folded hypercube

Weighted folded hypercube is an charming variance of the famous hypercube and is superior to the weighted hypercube in many criteria. We mainly study the scaling of network coherence for the weighted folded hypercube that is controlled by a weight factor. Network coherence quantifies the steady-state variance of these fluctuations, and it can be regarded as a measure of robustness of the consensus process to the additive noise. If networks with small steady-state variance have better network coherence, it can be regarded as more robust to noise than networks with low coherence. We firstly calculate the spectra of weighted folded hypercube and obtain the leading terms of network coherence that are quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. Finally, the results show that network coherence depends on iterations and weight factor. Meanwhile, with larger order, the scatings of the first- and second-order network coherence of weighted folded hypercube decrease with the increasing of weight factor.