The Smallest Values of Algebraic Connectivity for Trees

The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let Fn be the set of all trees of order n. In this paper, we will provide the ordering of trees in 3n up to the last eight trees according to their smallest algebraic connectivities when n ≥ 13. This extends the result of Shao et al. [The ordering of trees and connected graphs by algebraic connectivity. Linear Algebra Appl., 428, 1421-1438 （2008）].

Reducing power grid cascading failure propagation by minimizing algebraic connectivity in edge addition

Analyzing network robustness under various circumstances is generally regarded as a challenging problem.Robustness against failure is one of the essential properties of large-scale dynamic network systems such as power grids,transportation systems,communication systems,and computer networks.Due to the network diversity and complexity,many topological features have been proposed to capture specific system properties.For power grids,a popular process for improving a network’s structural robustness is via the topology design.However,most of existing methods focus on localized network metrics,such as node connectivity and edge connectivity,which do not encompass a global perspective of cascading propagation in a power grid.In this paper,we use an informative global metric algebraic connectivity because it is sensitive to the connectedness in a broader spectrum of graphs.Our process involves decreasing the average propagation in a power grid by minimizing the increase in its algebraic connectivity.We propose a topology-based greedy strategy to optimize the robustness of the power grid.To evaluate the network robustness,we calculate the average propagation using MATCASC to simulate cascading line outages in power grids.Experimental results illustrate that our proposed method outperforms existing techniques.

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.

A Lower Bound for the Algebraic Connectivity of a Graph in Terms of the Domination Number

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order n with fixed domination number γ≤（n＋2）/3, and finally present a lower bound for the algebraic connectivity in terms of the domination number. We also characterize the minimum algebraic connectivity of graphs with domination number half their order.

Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks

In this paper, the relations of the network topology and the moving consensus of multi-agent systems are studied. A consensus-prestissimo scale-free network model with the static preferential-consensus attachment is presented on the rewired link of the regular network. The effects of the static preferential-consensus BA network on the algebraic connectivity of the topology graph are compared with the regular network. The robustness gain to delay is analyzed for variable network topology with the same scale. The time to reach the consensus is studied for the dynamic network with and without communication delays. By applying the computer simulations, it is validated that the speed of the convergence of multi-agent systems can be greatly improved in the preferential-consensus BA network model with different configuration.

Topology Adaptation for Robust Ad Hoc Cyberphysical Networks under Puncture-Style Attacks

Many cyber physical networks will involve ad hoc deployments utilizing peer-to-peer communications. Examples include transportation systems where a group of moving cars communicate in order to avoid collisions, teams of robotic agents that work together in support of disaster recovery, and sensor networks deployed for health-care monitoring, monitoring the operation of a factory plant or to coordinate and actuate mechanisms for energy conservation in a building. These networks may face a variety of threats that puncture their connectivity and, should their performance degrade, the result could be catastrophic. Consider, for example, a vehicular ad hoc network where communication assists collision avoidance. In such a case, degradation could lead to vehicle accidents. Therefore, in order to overcome network performance degradations and the puncture of a network （such as blackhole or jamming） which is under attack, we propose an algorithm called the Fiedler Value Power Adjustment Topology Adaption （FVPATA）. FVPATA aims to dynamically adapt an ad hoc network＇s topology, even if the attacker varies its location and in the case of an interference-style attack by increasing the interference power. The algorithm utilizes the formulation from the graph theory which works with the Fiedler value to guide each node in wireless ad hoc network utilizing power adjustments to enhance the network＇s overall robustness. The advantage of the proposed mechanism is that it is a light-weight approach which is totally distributed, based on topology updates inherent in the Optimized Link State Routing （OLSR） protocol and, hence, it is unnecessary to introduce additional messages. Additionally, an algorithm was developed to resolve problems involving asymmetric links that arise in ad hoc networks by eliminating unnecessary energy consumption of Fiedler nodes. Simulation results using NS3 show that the proposed mechanism successfully decreases the average amount of hops used by 50% and the delay of flows when nodes are migrating at a modest rate below 60 m/min.

On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an alge- braically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group Hi （A） is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Some Characterizations of Algebras of Finite Representation Type

Let A be a finite dimensional, connected, basic algebra over an algebraically closed field. We prove that A is of finite representation type if and only if there is a natural number m such that rad^m（End（M）） = 0, for any indecomposable A-modules M. This gives a partial answer to one of problems posed by Skowrofiski.