Asymptotic energy of graphs

The energy of a simple graph $G$ arising in chemical physics, denoted by $\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if $\{G_n\}$ is a sequence of finite simple graphs with bounded average degree and $\{G_n'\}$ a sequence of spanning subgraphs of $\{G_n\}$ such that almost all vertices of $G_n$ and $G_n'$ have the same degrees, then $G_n$ and $G_n'$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.Comment: 15 pages, 3 figure

Extremal Matching Energy of Complements of Trees

Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let $T$ be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . ., \floor{ n/2 } . When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy

Extremal Matching Energy of Complements of Trees

Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . . , [n/2]. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy

Delay-Dependent Stability Criteria of Uncertain Periodic Switched Recurrent Neural Networks with Time-Varying Delays

This paper deals with the problem of delay-dependent stability criterion of uncertain periodic switched recurrent neural networks with time-varying delays. When uncertain discrete-time recurrent neural network is a periodic system, it is expressed as switched neural network for the finite switching state. Based on the switched quadratic Lyapunov functional approach (SQLF) and free-weighting matrix approach (FWM), some linear matrix inequality criteria are found to guarantee the delay-dependent asymptotical stability of these systems. Two examples illustrate the exactness of the proposed criteria

Design of a UHF Antenna for Partial Discharge Detection of Power Equipment

A single-arm Archimedean spiral antenna that can be directly fed by a 50 Ω coaxial cable is investigated in this study. Every antenna parameter is optimized under simulation to make the antenna work in the ultra-high frequency band. The influence of dielectric materials, feed cone angle, and antenna duty ratio is also examined. Partial discharge (PD) experiments on several typical artificial insulation defects are conducted, and a single-arm Archimedean spiral antenna and a typical microstrip antenna are utilized for PD measurement. The PD characteristics of different insulation defects are also analyzed. Results show that the designed antenna is suitable for ultra-high frequency monitoring. The detection sensitivity of the single-arm spiral antenna is superior to that of the ordinary microstrip antenna. The former can be utilized in wide-band measurement fields

Delay-Dependent Stability Criterion of Arbitrary Switched Linear Systems with Time-Varying Delay

This paper deals with the problem of delay-dependent stability criterion of arbitrary switched linear systems with time-varying delay. Based on switched quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criterions are found to guarantee delay-dependent asymptotically stability of these systems. Simultaneously, arbitrary switched linear system can be expressed as a problem of uncertain liner system, so some delay-dependent stability criterions are obtained with the result of uncertain liner system. Two examples illustrate the exactness of the proposed criterions

Delay-dependent stability criterion of arbitrary switched linear systems with time-varying delay,”

This paper deals with the problem of delay-dependent stability criterion of arbitrary switched linear systems with time-varying delay. Based on switched quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criterions are found to guarantee delay-dependent asymptotically stability of these systems. Simultaneously, arbitrary switched linear system can be expressed as a problem of uncertain liner system, so some delaydependent stability criterions are obtained with the result of uncertain liner system. Two examples illustrate the exactness of the proposed criterions

Detection of Water Content in Transformer Oil Using Multi Frequency Ultrasonic with PCA-GA-BPNN

The water content in oil is closely related to the deterioration performance of an insulation system, and accurate prediction of water content in oil is important for the stability and security level of power systems. A novel method of measuring water content in transformer oil using multi frequency ultrasonic with a back propagation neural network that was optimized by principal component analysis and genetic algorithm (PCA-GA-BPNN), is reported in this paper. 160 oil samples of different water content were investigated using the multi frequency ultrasonic detection technology. Then the multi frequency ultrasonic data were preprocessed using principal component analysis (PCA), which was implemented to obtain main principal components containing 95% of original information. After that, a genetic algorithm (GA) was incorporated to optimize the parameters for a back propagation neural network (BPNN), including the weight and threshold. Finally, the BPNN model with the optimized parameters was trained with a random 150 sets of pretreatment data, and the generalization ability of the model was tested with the remaining 10 sets. The mean squared error of the test sets was 8.65 × 10−5, with a correlation coefficient of 0.98. Results show that the developed PCA-GA-BPNN model is robust and enables accurate prediction of a water content in transformer oil using multi frequency ultrasonic technology