These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate ...

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G − F contains at least two vertices. LetGi be a connected graph with order ni , minimum degree δi and edge-connectivity λi for i = 1, 2. This article shows that λ′(G1 × G2) ≥ min{n1 λ2,n2 λ1,λ1 + 2λ2, 2λ1+λ2} forn1,n2 ≥ 3 andλ′(K2×G2) = min{n2, 2λ2}, which general...

One of the promising lightwave network architectures is the multihop lightwave networks based on wavelength division multiplexing (WDM). In this architecture, separate channels created by assigning different wavelengths between node pairs define the logical connectivity. This logical topology is independent of the underlying physical topology and can be adaptively changed according to the chang...

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain (k− 1)-connected for up to ⌈t/k⌉ rounds, where t is the maximum number of Byzantine processes, and t ≥ k ≥ 1. Protocol complex connectivity is important since a (k − 1)-connected protocol complex does not admit solutions for the k-set agreement problem. In crash-failure systems, the connectivity upper ...

Coral reefs are increasingly exposed to elevated temperatures that can cause coral bleaching and high levels of mortality of corals and associated organisms. The temperature threshold for coral bleaching depends on the acclimation and adaptation of corals to the local maximum temperature regime. However, because of larval dispersal, coral populations can receive larvae from corals that are adap...

in this paper we discuss about tenacity and its properties in stability calculation. we indicate relationships between tenacity and connectivity, tenacity and binding number, tenacity and toughness. we also give good lower and upper bounds for tenacity.

The right and left hand Motor Imagery (MI) analysis based on the electroencephalogram (EEG) signal can directly link the central nervous system to a computer or a device. This study aims to identify a set of robust and nonlinear effective brain connectivity features quantified by transfer entropy (TE) to characterize the relationship between brain regions from EEG signals and create a hierarchi...

It has been shown [Van den Broeck and Kawai, 1990, Patarnello and Carnevali, 1987] that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transf...

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v1, . . . , vk of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k-orderedness of a graph implies (k − 1)-connec...

Multi-connectivity is considered to be key for enabling reliable transmissions and enhancing data rates in future wireless networks. In this work, we quantify the communication performance by the outage probability and the system throughput. We establish a remarkably simple, yet accurate analytical framework based on distributed source coding to describe the outage probability and the system th...