Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

3-extraconnectivity of Cayley Graphs Generated by Transposition Generating Trees

给定一个图g和一个非负整数g,若图g中存在(边)点集,使得删除该集合后图g不连通并且每个连通分支的点数大于g,所有这样的(边)点集的最小基数,称为g-额外(边)连通度(记作κg(g)(λg(g)).本文将确定由对换树生成的凯莱图的3-额外(边)连通度(记作κ3(λ3).Given a graph G and a non-negative integer g, the g-extra(edge) connectivity of G (written κg(G)(λg(G))is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices.In this paper, we determine 3-extra(edge) connectivity(written κ3(λ3)) of Cayley graphs generated by transposition trees.supportedbyNSFC(No.10671165

Optical Wireless Data Center Networks

Bandwidth and computation-intensive Big Data applications in disciplines like social media, bio- and nano-informatics, Internet-of-Things (IoT), and real-time analytics, are pushing existing access and core (backbone) networks as well as Data Center Networks (DCNs) to their limits. Next generation DCNs must support continuously increasing network traffic while satisfying minimum performance requirements of latency, reliability, flexibility and scalability. Therefore, a larger number of cables (i.e., copper-cables and fiber optics) may be required in conventional wired DCNs. In addition to limiting the possible topologies, large number of cables may result into design and development problems related to wire ducting and maintenance, heat dissipation, and power consumption. To address the cabling complexity in wired DCNs, we propose OWCells, a class of optical wireless cellular data center network architectures in which fixed line of sight (LOS) optical wireless communication (OWC) links are used to connect the racks arranged in regular polygonal topologies. We present the OWCell DCN architecture, develop its theoretical underpinnings, and investigate routing protocols and OWC transceiver design. To realize a fully wireless DCN, servers in racks must also be connected using OWC links. There is, however, a difficulty of connecting multiple adjacent network components, such as servers in a rack, using point-to-point LOS links. To overcome this problem, we propose and validate the feasibility of an FSO-Bus to connect multiple adjacent network components using NLOS point-to-point OWC links. Finally, to complete the design of the OWC transceiver, we develop a new class of strictly and rearrangeably non-blocking multicast optical switches in which multicast is performed efficiently at the physical optical (lower) layer rather than upper layers (e.g., application layer). Advisors: Jitender S. Deogun and Dennis R. Alexande

Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states

Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity and is closely related to Matchgate circuits introduced by Valiant. Mathematically, these classes of circuits can be understood as fermionic representations of the Lie groups $U(d)$ and $SO(2d)$. This observation allows us to prove our main technical results. We first show anticoncentration for probabilities in random FLO circuits of both kind. Moreover, we prove robust average-case hardness of computation of probabilities. To achieve this, we adapt the worst-to-average-case reduction based on Cayley transform, introduced recently by Movassagh, to representations of low-dimensional Lie groups. Taken together, these findings provide hardness guarantees comparable to the paradigm of Random Circuit Sampling. Importantly, our scheme has also a potential for experimental realization. Both passive and active FLO circuits are relevant for quantum chemistry and many-body physics and have been already implemented in proof-of-principle experiments with superconducting qubit architectures. Preparation of the desired quantum input states can be obtained by a simple quantum circuit acting independently on disjoint blocks of four qubits and using 3 entangling gates per block. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified.Comment: 65 pages, 13 figures, 1 table, v2: improved discussion and narrative, numerics about anticoncentration added, references updated, comments and suggestions are welcom