An extremal problem on group connectivity of graphs

Let A be an Abelian group, n \u3e 3 be an integer, and ex(n, A) be the maximum integer such that every n-vertex simple graph with at most ex(n, A) edges is not A-connected. In this paper, we study ex(n, A) for IAI \u3e 3 and present lower and upper bounds for 3 \u3c IAI 5. 0 2012 Elsevier Ltd. All rights reserved

Hamilton-chain saturated hypergraphs

AbstractWe say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamiltonian path (cycle) saturated hypergraph, mainly for r=3. We present a construction of a family of 3-uniform path (cycle) saturated hamiltonian hypergraphs with O(n5/2) edges. On the other hand we prove that the number of edges in an r-uniform hamiltonian path (cycle) saturated hypergraph is at least Ω(nr−1)

Integer flows and Modulo Orientations

Tutte\u27s 3-flow conjecture (1970\u27s) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A graph G admits a nowhere-zero 3-flow if and only if G has an orientation such that the out-degree equals the in-degree modulo 3 for every vertex. In the 1980ies Jaeger suggested some related conjectures. The generalized conjecture to modulo k-orientations, called circular flow conjecture, says that, for every odd natural number k, every (2k-2)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex. And the weaker conjecture he made, known as the weak 3-flow conjecture where he suggests that the constant 4 is replaced by any larger constant.;The weak version of the circular flow conjecture and the weak 3-flow conjecture are verified by Thomassen (JCTB 2012) recently. He proved that, for every odd natural number k, every (2k 2 + k)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex and for k = 3 the edge-connectivity 8 suffices. Those proofs are refined in this paper to give the same conclusions for 9 k-edge-connected graphs and for 6-edge-connected graphs when k = 3 respectively

Modeling and Tuning of Energy Harvesting Device Using Piezoelectric Cantilever Array

Piezoelectric devices have been increasingly investigated as a means of converting ambient vibrations into electrical energy that can be stored and used to power other devices, such as the sensors/actuators, micro-electro-mechanical systems (MEMS) devices, and microprocessor units etc. The objective of this work was to design, fabricate, and test a piezoelectric device to harvest as much power as possible from vibration sources and effectively store the power in a battery.;The main factors determining the amount of collectable power of a single piezoelectric cantilever are its resonant frequency, operation mode and resistive load in the charging circuit. A proof mass was used to adjust the resonant frequency and operation mode of a piezoelectric cantilever by moving the mass along the cantilever. Due to the tiny amount of collected power, a capacitor was suggested in the charging circuit as an intermediate station. To harvest sufficient energy, a piezoelectric cantilever array, which integrates multiple cantilevers in parallel connection, was investigated.;In the past, most prior research has focused on the theoretical analysis of power generation instead of storing generated power in a physical device. In this research, a commercial solid-state battery was used to store the power collected by the proposed piezoelectric cantilever array. The time required to charge the battery up to 80% capacity using a constant power supply was 970 s. It took about 2400 s for the piezoelectric array to complete the same task. Other than harvesting energy from sinusoidal waveforms, a vibration source that emulates a real environment was also studied. In this research the response of a bridge-vehicle system was used as the vibration sources such a scenario is much closer to a real environment compared with typical lab setups