Let G be a graph with vertex set V ( G ) and edge set E ( G ) . The mixed metric dimension of a connected graph G , denoted by dim m ( G ) , is the minimum cardinality of a subset S ⊆ V ( G ) such that for any two u , v ∈ V ( G ) ∪ E ( G ) , there exists w ∈ S so that the distance between w and u is not equal to the distance between w and v . In this paper, we present further results on the mixed metric dimension. First, we give a sharp upper bound on the mixed metric dimension for a graph in terms of the number of cut vertices of this graph. Second, we compare the mixed metric dimension with geodesic transversal number for trees , unicyclic graphs and block graphs. Finally, we provide some new results about a conjecture, due to Sedlar and Škrekovski (Sedlar and Škrekovski, 2021), on the mixed metric dimension.

A graph G is a fractional (a,b,n)-critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional [a,b]-factor. In this paper, we determine the exact tight isolated toughness bound for fractional (a,b,n)-critical graphs. To be specific, a graph G is fractional (a,b,n)-critical if δ(G)≥a+n and I(G)>a−1+n+1na,b, where na,b≥2 is an integer satisfies (na,b−1)a≤b≤na,ba−1. Furthermore, the sharpness of bounds is showcased by counterexamples. Our contribution improves a result from [W. Gao, W. Wang, and Y. Chen, Tight isolated toughness bound for fractional (k,n)-critical graphs, Discrete Appl. Math. 322 (2022), 194–202] which established the tight isolated toughness bound for fractional (k,n)-critical graphs.

Recent studies have shown that there is a profound connection between the existence of H -factors under attack circumstances and the parameters to measure the vulnerability of the network. The disadvantage of the previous theoretical conclusions is that the connectivity is often regarded as fixed, and the relationship between other network parameters and the H -factor is explored. However, in a real situation, as the network structure changes, connectivity becomes a dynamically variational parameter, and its changing will affect other parameters to make corresponding changes. In this study, we treat all parameters as a complex system that restricts each other, consider their mutual constraints from a geometric point of view, and apply high-dimensional surfaces to characterize their relationships. Seven network parameters including toughness and binding number are considered, and the concrete expression form of the surfaces in several specific settings are obtained. Furthermore, the local version for these seven network parameters are introduced.

Isolation toughness is a vital parameter to evaluate the vulnerability of computer networks. In specific network designing stage, it is necessary to find the lower bound of the isolated toughness, and strive to build a network that meets the stability requirements with the least cost. Gao et al.1 conjectured that if a graph G with κ ( G ) ≥ 3 m + 1 2 satisfies I ( G ) > 7 m + 5 4 m + 4 or I ′ ( G ) > 7 m + 5 4 m + 2 , then G is a ( P ≥ 3 , m ) -factor deleted graph. It's proved that this conjecture holds. However, it is found that as the connectivity changes, the tight lower bound of isolated toughness for ( P ≥ 3 , m ) -factor deleted graphs will change as well. Therefore, we propose a new perspective to look into this problem and introduce the concepts of isolated toughness ( P ≥ 3 , m ) factor deleted surface and isolated toughness variant ( P ≥ 3 , m ) factor deleted surface, where the result of the original conjecture is only a cross-section on surfaces. The main contribution in this paper is to determine the concrete expression of these two surfaces.

The issues of ruggedness and vulnerability are cruxes in network security research, which must be considered during the network designing phase. Parameters such as toughness, isolated toughness, and binding number characterize the vulnerable of the network from the structure of networks. The path factor, a special case of the generalized ℋ -factor, measures the feasibility of data transmission in networks. Recent advances have been obtained to show that there is an inevitable connection between the vulnerability parameters of the network and the existence of path factors, while we found that some existing theoretical results are not tight and there is still a long way for further improvement. In view of graph theory approaches, this paper mainly contributes to determine the sharp bounds of toughness, isolated toughness, and binding number for the existence of path factor in different settings, and therefore solve the open problems left unsolved in previous articles.

In photovoltaic power generation, UPS and other applications, DC/AC converters with a low input DC voltage are essentially used. However, this type of converter is commonly put to use the traditional two-level converting topology, which results problems of high harmonic content in the output voltage and large power loss. Therefore, a single stage multilevel converter based on transformer multi-tap voltages control fed by a low DC voltage source (MCMTVC) is proposed in this paper. The converter achieves multi levels by control different transformer tap voltages through the power switches connected to the taps, and the number of output levels can be conveniently expanded through the expansion of the transformer taps. During operation, the power switch branches that each branch is connected in series by at most two power transistors including power diodes take turns to serve power exchange, and the voltage stress upon the power switches active in PWM state is very low. The principle of the converter is described in this paper in the first, and then the corresponding control method is put forward, furthermore the characteristics of the converter are discussed. In final, A 9-level simulation platform with the mentioned topology and control strategy using the MATLAB / Simulink software is established, and then simulation research is carried out, whose results have verified the proposed control strategy and conclusions.

This paper studies and proposes a high-efficiency method for linear precision AC voltage regulation and filtering, which can increase the load-carrying capacity and load adaptability of the voltage-regulated output while maintaining the same filtering effect. Taking the emitter-output linear circuit as the basic research object, the single-stage linear precision AC voltage regulation and filter circuit topology is first proposed. Then on this basis, on the premise of the voltage regulation efficiency, the voltage regulation range is further expanded. A multi-stage linear voltage regulation structure is given, and the distribution law of its efficiency with voltage change is obtained through simulation. While retaining the advantages of the traditional linear amplifier with high waveform quality and no high-frequency noise, it greatly improves system efficiency. Finally, simulations and experiments verify the correctness of this topology and accuracy of the voltage regulation and filtering methods.

In medium and high voltage applications, hybrid active power filter (HAPF) play an important role among harmonic elimination methods. How to connect active compensator into high-voltage grid is a hot research topic. In this paper, a hybrid active power filter based on coupling inductor is proposed, where the passive filters are applied to filter out most harmonics, and the active filter is used to compensate the residual characteristic harmonics causing by tuning deviation of the passive filters and other harmonics. Due to the coupling inductor, the self-inductance is generally small, which limits possible high voltage at both sides of the coupling inductor. In order to obtain enough voltage for the capacitor at the dc side of the active converter to generate required compensating currents, a normalized signal of the coupling voltage is used as the unit active reference signal, so that all frequency components of the coupling voltage source can provide energy for the dc capacitor. For harmonic compensating a control strategy with a unified proportion controller and some auxiliary characteristic harmonic resonance controllers is presented. A comparative study of filtering effects by different filtering methods is carried out through simulation, and then does the control characteristics of the active converter, which shows super performance.

Network virtualization provides a promising solution to overcome the ossification of current networks, allowing multiple Virtual Network Requests (VNRs) embedded on a common infrastructure. The major challenge in network virtualization is the Virtual Network Embedding (VNE) problem, which is to embed VNRs onto a shared substrate network and known to be NP-hard. The topological heterogeneity of VNRs is one important factor hampering the performance of the VNE. However, in many specialized applications and infrastructures, VNRs are of some common structural features e.g., paths and cycles. To achieve better outcomes, it is thus critical to design dedicated algorithms for these applications and infrastructures by taking into accounting topological characteristics. Besides, paths and cycles are two of the most fundamental topologies that all network structures consist of. Exploiting the characteristics of path and cycle embeddings is vital to tackle the general VNE problem. In this paper, we investigated the path and cycle embedding problems. For path embedding, we utilize Multiple Knapsack Problem (MKP) and Multi-Dimensional Knapsack Problem (MDKP), we proposed an efficient and effective MKPMDKP-based algorithm. For cycle embedding, we proposed a Weighted Directed Auxiliary Graph (WDAG) to develop a polynomial-time algorithm to determine the least-resourceconsuming embedding. Numerical results showed our customized algorithms can boost the acceptance ratio and revenue compared to generic embedding algorithms in the literature.

Elastic Optical Network (EON) has been considered as a promising optical networking technology to architect the next-generation backbone networks. The spectrum management in EONs is directly determined by the Routing and Spectrum Assignment (RSA). Generally, the RSA is solved by routing the requests with lightpaths first and then assigning spectrum resources to the lightpaths to optimize the spectrum usage. Thus, the spectrum assignment explicitly determines the spectrum usage. Besides, the network topology, traffic distribution and routing scheme implicitly impact the spectrum usage. However, few related work involves this implicit impact. In this paper, we aim to provide a thoroughly theoretical analysis on the impact of the three key factors on the spectrum usage. To this end, two theoretical chains are proposed: (1) The optimal spectrum usage can be measured by the chromatic number of the conflict graph, which is positively correlated to the intersecting probability, i.e., the smaller the intersecting probability, the smaller the optimal spectrum usage; (2) The intersecting probability is determined by the network topology, traffic distribution and routing scheme via a quadratic programming parameterized with a matrix of conflict coefficients. The effectiveness of our theoretical analysis has been validated by extensive numerical results.

In elastic optical networks, two lightpaths sharing common fiber links might have to be isolated in the spectrum domain with a proper guard-band to prevent crosstalk and/or reduce physical-layer security threats. Meanwhile, the actual requirements on guard-band sizes can vary for different lightpath pairs, because of various reasons. Therefore, in this paper, we consider the situation in which the actual guardband requirements for different lightpath pairs are different, and formulate the distance spectrum assignment (DSA) problem to investigate how to assign the spectrum resources efficiently in such a situation. We first define the DSA problem formally and prove its NP-hardness and inapproximability. Then, we analyze and provide the upper and lower bounds for the optimal solution of DSA, and prove that they are tight. In order to solve the DSA problem time-efficiently, we develop a two-phase algorithm. In its first phase, we obtain an initial solution and then the second phase improves the quality of the initial solution with random optimization. We prove that the proposed two-phase algorithm can get the optimal solution in bipartite DSA conflict graphs and can ensure an approximate ratio of O(log(|V|)) in complete DSA conflict graphs, where |V| is the number of vertices in the conflict graph, i.e., the number of lightpaths to be considered. Numerical results demonstrate our proposed algorithm can find near-optimal solutions for DSA in various conflict graphs.