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Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, in which two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let $K_{k,n-k}$ be a complete bipartite graph of order $n$. In 1974, Wilson characterized the connectedness of $\mathsf{FS}(K_{1,n-1},Y)$ by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of $\mathsf{FS}(K_{k,n-k},Y)$ for any $Y$ and all $k\ge 2$, including $Y$ being a random graph, as suggested by Defant and Kravitz, and pose some open problems.

A distributed zero-sequence-current (ZSC) carrier signal injection via current transformer (CT) is proposed for underground cable single-phase-to-ground fault location in neutral isolated medium voltage (MV) distribution networks. High-frequency ZSC carrier signal is directly injected to a high-voltage underground cable of an energized distribution network via CT. The injected carrier signal travels along the cable to the receiving center in the substation, and is processed for quick fault line selection and fault location. No external power supplies or signal generators are needed due to the self-contained nature of the injector. The device can be constructed as a sleeve to be mounted around the underground power cable. This permits economical retrofitting to existing installations of power line communication (PLC) devices. In this paper, the operational principle and the application considerations for such an injection device and fault location tricks are explained, followed by the theory behind the proposed technique. Finally, MATLAB based numerical simulations are performed and a low-voltage prototype is tested to verify the validity of the proposed approach.

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k , the Gallai Ramsey number G R k ( H ) is the minimum integer N such that every Gallai k -coloring of K N contains a monochromatic copy of H . A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and l , the l -uniform Ramsey number R k l ( H ) is the minimum integer N such that any uniform k -coloring of any complete multipartite graph on N vertices with each part of cardinality no more than l , contains a monochromatic copy of H . Let K m , n denote a complete bipartite graph. Wu et al. conjectured that G R k ( K m , n ) = ( n − 1 ) ( k − 2 ) + R 2 1 ( K m , n ) if R 2 1 ( K m , n ) ≥ 3 m − 2 , where k ≥ 3 and m ≥ n ≥ 2 . In this paper, we show that G R k ( K m , n ) = ( n − 1 ) ( k − 2 ) + R 2 m − 1 ( K m , n ) for all integers k ≥ 3 and m ≥ n ≥ 2 . Based on this result, we improve the known general bounds for G R k ( K m , n ) , and confirm the conjecture for some complete bipartite graphs.

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k -coloring is a Gallai coloring that uses at most k colors. Given an integer k ≥ 2 , and graphs H 1 , H 2 , … , H k , the Gallai-Ramsey number G R ( H 1 , H 2 , … , H k ) is the least positive integer N such that every Gallai k -coloring of the complete graph K N contains a monochromatic copy of H i in color i for some i ∈ { 1 , 2 , … , k } . Let m G denote the union of m disjoint copies of G . This paper studies the Gallai–Ramsey number G R ( t 1 K 3 , … , t k K 3 ) for all integers t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1 . The case t 1 = t 2 = ⋯ = t k = 1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of G R ( t 1 K 3 , … , t k K 3 ) for all t 1 ≥ t 2 ≥ ⋯ ≥ t k ≥ 1 , and prove that both the upper and lower bounds can be attained for some t i , i ∈ { 1 , … , k } .

For two graphs $$G_1$$ and $$G_2$$ , the star-critical Ramsey number $$r_*(G_1,G_2)$$ is the minimum integer k such that any red/blue edge-coloring of $$K_{r-1}\sqcup K_{1,k}$$ contains a red copy of $$G_1$$ or a blue copy of $$G_2$$ , where r is the classical Ramsey number $$R(G_1,G_2)$$ and $$K_{r-1}\sqcup K_{1,k}$$ is the graph obtained from a $$K_{r-1}$$ and an additional vertex v by joining v to k vertices of $$K_{r-1}$$ . Let $$C_n$$ denote a cycle of order n and $$W_m$$ a wheel of order $$m+1$$ . Hook (2010) proved that $$r_*(C_n,W_3)=2n$$ for $$n\ge 5$$ . In this paper, it is shown that $$r_*(C_n,W_m)=2n$$ for m odd, $$n\ge m\ge 5$$ and $$n\ge 60$$ .

The generalized Turan number $$ex(n,K_s,H)$$ is defined to be the maximum number of copies of a complete graph $$K_s$$ in any H-free graph on n vertices. Let F be an even linear forest, that is, F consists of paths of even order. In this paper, we determine the value of $$ex(n,K_s,F)$$ , which generalizes or improves some known results on generalized Turan number and classical Turan number.

In this article, the random vibration test and finite-element simulation analysis of SAC305 solder joints in BGA packaged electronic chips were carried out. A failure model for BGA solder joints was established based on the life data. And vulnerable parts in electronic chips were analyzed. First, the random vibration test was conducted. Second, a lognormal distribution model of solder joints by means of the life data was established and analyzed based on the fracture mechanics theory. Finally, the finite-element simulation model of BGA packaged electronic chip was established and discussed. Failure characteristics of solder joints at different positions in electronic chips were compared and the failure mode of the solder joint was discussed using the simulation result and the observation result with scanning electron microscope (SEM). The results show that the life of each single solder joint in BGA packaged electronic chips follows a logarithmic normal distribution under the random vibration. Solder balls located at the four outermost corners of the chip are subjected to the greatest stress, which are the weakest parts in electronic chips. Cracks in solder joints are generated near the package side and emerged from the edge of the intermetallic compound (IMC) layer with the largest curvature.

Objective. This study aims to find the novel lncRNAs closely related to the progression of oral squamous cell carcinoma (OSCC) by comprehensively analyzing microarray. Methods. Chip dataset GSE84805 was downloaded from the Gene Expression Omnibus (GEO) database, lncRNA expression profiles of OSCC and paracancerous tissue were obtained, probes sequences reannotation was conducted, and differentially expressed lncRNAs (DELs) and differentially expressed genes (DEGs) were identified. Finally, these data were analyzed by constructing the lncRNA-function enrichment network. Results. We found that 465 lncRNAs are differentially expressed consisting of 193 upregulated lncRNAs and 272 downregulated lncRNAs. Meanwhile, 811 DEGs were identified with 498 upregulated genes and 313 downregulated genes. Analysis of the lncRNA-function enrichment network showed that these aberrant lncRNAs may be related to focal adhesion, inflammatory response pathway, cell cycle, matrix metalloproteinases, and other biological functions. Also, we found that some key lncRNAs such as LINC00152 and HOXA11-AS have been shown to play an important role in tumor proliferation and migration. Conclusion. The key lncRNAs may serve as potential molecular markers or therapeutic targets in OSCC formation and development. It can also help us to understand the molecular mechanism of occurrence and development in OSCC.

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$, of a graph $G$, is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$. Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$. In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$.

Let $$G=(V(G),E(G))$$ be a graph and s, t integers with $$s\le t$$. If we can assign an s-subset $$\phi (v)$$ of the set $$\{1, 2,\ldots ,t\}$$ to each vertex v of V(G) such that $$\phi (u)\cap \phi (v)=\emptyset $$ for every edge $$uv\in E(G)$$, then G is called (t : s)-colorable, and such an assignment $$\phi $$ is called a (t : s)-coloring of G. Let $$C_n$$ denote a cycle of length n. In this paper, we show that every planar graph without $$C_4$$ and $$C_6$$ is (7 : 2)-colorable and thus has fractional chromatic number at most 7/2.

Solenoid valves are important electromagnetic devices which are used widely in various fields. Considering the uncertainty of failure mechanisms in practical operation, a thermal-structure coupled model of a direct action solenoid valve is constructed by using the finite element method. The coupled model provides useful information such as temperature distribution and stress distribution for failure mechanism study of the solenoid valve. The model results predict that the failure of the solenoid valve is closely related to the thermal expansion inside the coil, which means the high temperature generated by thermal expansion melts the insulation layers. Under the combined action of high temperature and coupled stress, the shorting of coils occurs and the resistance decreases which may cause the eventual failure of the solenoid valve. To verify the prediction of the finite element model, a degradation experiment is designed. The results suggest that the failure of the solenoid valve is a gradual degradation process, and there are several mutations of coil temperature and resistance. The high temperature melts the insulation layers and causes the shorting of coils, which further decreases the resistance and increases the current. Later more heat is generated to rise coil temperature and melt insulation layers, which will short out the coils and drop the resistance. With such a relationship, as the temperature rises alternately, the resistance decreases alternately until the solenoid valve completely fails. The model provides a reference for failure mechanism study and life prolonging research of solenoid valves.

Consider two graphs $X$ and $Y$, each with $n$ vertices. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, where two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Alon, Defant, and Kravitz showed that if $X$ and $Y$ are two independent random graphs in $\mathcal{G}(n,p)$, then the threshold probability guaranteeing the connectedness of $\mathsf{FS}(X,Y)$ is $p_0=n^{-1/2+o(1)}$, and suggested to investigate the general asymmetric situation, that is, $X\in \mathcal{G}(n,p_1)$ and $Y\in \mathcal{G}(n,p_2)$. In this paper, we show that if $p_1 p_2 \ge p_0^2=n^{-1+o(1)}$ and $p_1, p_2 \ge w(n) p_0$, where $w(n)\rightarrow 0$ as $n\rightarrow \infty$, then $\mathsf{FS}(X,Y)$ is connected with high probability, which extends the result on $p_1=p_2=p$, due to Alon, Defant, and Kravitz., Comment: 16 pages, 1 fighres. This is a version revised mainly under very careful comments from the anonymous referees. arXiv admin note: substantial text overlap with arXiv:2009.07840 by other authors

For two given graphs $$G_1$$ and $$G_2$$, the planar Ramsey number $$PR( G_1,G_2)$$ is the smallest integer N such that every planar graph G on N vertices either contains a copy of $$G_1$$ or its complement contains a copy of $$G_2$$. In this paper, we determine all planar Ramsey numbers for the complete graphs versus trees.

For k given graphs G 1 , G 2 , … , G k , k ≥ 2 , the k -color Ramsey number, denoted by R ( G 1 , G 2 , … , G k ) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of G i colored with i , for some 1 ≤ i ≤ k . Let C m be a cycle of length m and K 1 , n a star of order n + 1 . In this paper, firstly we give a general upper bound of R ( C 4 , C 4 , … , C 4 , K 1 , n ) . In particular, for the 3-color case, we have R ( C 4 , C 4 , K 1 , n ) ≤ n + 4 n + 5 + 3 and this bound is tight in some sense. Furthermore, we prove that R ( C 4 , C 4 , K 1 , n ) ≤ n + 4 n + 5 + 2 for all n = l 2 − l and l ≥ 2 , and if l is a prime power, then the equality holds.

The unbalance current issue of the high-speed railway is the most prominent power quality of the traction power supply systems. Bullet trains can generate energy feedback when they brake, which should not be neglected. In this study, the effects of energy feedback on the unbalance issue are analysed, under different operating conditions of the trains. A collaborative unbalance compensation method (CUCM) for the high-speed railway is proposed in this work to lower the compensation capacity of the expensive power electronics compensation devices. To ensure that the CUCM works well, a further analysis of the two compensation principles, the railway power conditioner principle and Steinmetz principle, was carried out. The analysis results showed that the two compensation principles have different implementation methods and compensation current expressions, but the compensation essences were the same. According to the principle analysis results, a collaborative strategy was used. A capacity analysis of the CUCM, based on the actual operating situations of the bullet trains, was also completely done in this study. The simulation and experimental results showed that the proposed CUCM could adequately compensate the unbalance current of the high-speed railway power systems well with a low cost, in both traction and energy feedback situations.

For given graphs H 1 , . . . , H k , k ≥ 2 , the k -color Ramsey number R ( H 1 , . . . , H k ) is the smallest integer N such that every k -coloring of the edges of a complete graph K N contains a monochromatic copy of H i colored in i , for some i with 1 ≤ i ≤ k . Let C l , K 1 , m and P n denote a cycle of length l , a star of order m + 1 and a path of order n , respectively. In this paper, it is shown that R ( C 4 , K 1 , m , P n ) ≤ m + n − 1 + ⌈ m + n − 2 ⌉ for all m , n ≥ 2 and R ( C 4 , K 1 , m , P n ) ≤ m + n − 2 + ⌈ m + n − 2 ⌉ if m + n = l 2 + 3 and l ≥ 1 . Moreover, by discussing the local structure of the polarity graph whose vertices are points in the projective plane over Galois fields, we show that the two upper bounds can be attained for some special m and n . These results also extend some known results on R ( C 4 , K 1 , m ) obtained by Parsons in 1975 and by Zhang et al. recently.

In the networking designing phase, the network needs to be built according to certain indicators to ensure that the network has the ideal functions and can work smoothly. From a modeling perspective, each site in the network is represented by a vertex, channels between sites are represented by edges, and thus the entire network can be denoted as a graph. Problems in the network can be transformed into corresponding graph problems. In particular, the feasibility of data transmission can be transformed into the existence of fractional factors in network graph. This note gives an independent set neighborhood union condition for the existence of fractional factors in a special setting, and shows that the neighborhood union condition is sharp.

This paper proposes a hybrid active power filter (HAPF) with multi-coupled coils, applied to a medium- and high-voltage power grid. The passive filter of the proposed HAPF adopts the structure of multi-coupled coils to compress the traditional multiple LC branches into one branch which presents the same harmonic impedance characteristics as the former multiple LC branches. In the active power filter of the HAPF, a coupled inductor, instead of a transformer, is used to connect with the passive filter. The coupled inductor has mutual inductances with inductors of the passive filter. Through spatial magnetic coupling, the active power filter can inject compensation current into the power grid to eliminate the residual harmonics and absorb active power from the power grid to maintain the DC capacitor voltage. When the active power filter is open-circuited or short-circuited, the filtering effect of the passive filter can still be guaranteed, which improves the reliability of the filter. The benefits of the proposed HAPF with excellent harmonic filtering performance are that the inductors occupy only 1/3 space as compared with traditional three-tuned LC filter, and very small power of the active power filter. The feasibility of the proposed HAPF is verified through simulations and experiment.

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.

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 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.

In the past ten years, researchers have always attached great importance to the application of ontology to its relevant specific fields. At the same time, applying learning algorithms to many ontology algorithms is also a hot topic. For example, ontology learning technology and knowledge are used in the field of semantic retrieval and machine translation. The field of discovery and information systems can also be integrated with ontology learning techniques. Among several ontology learning tricks, multi-dividing ontology learning is the most popular one which proved to be in high efficiency for the similarity calculation of tree structure ontology. In this work, we study the multi-dividing ontology learning algorithm from the mathematical point of view, and an approximation conclusion is presented under the linear representation assumption. The theoretical result obtained here has constructive meaning for the similarity calculation and concrete engineering applications of tree-shaped ontologies. Finally, linear combination multi-dividing ontology learning is applied to university ontologies and mathematical ontologies, and the experimental results imply that the higher efficiency of the proposed approach in actual applications.

For two given graphs $$G_1$$ and $$G_2$$ , the Ramsey number $$R(G_1,G_2)$$ is the smallest integer n such that for any graph G of order n, either G contains $$G_1$$ or its complement $${\overline{G}}$$ contains $$G_2$$ . Let $$P_n, S_n$$ and $$T_n$$ denote a path, a star and a tree of order n, respectively. A generalized wheel, denoted by $$W_{s,m}$$ , is the join of a complete graph $$K_s$$ and a cycle $$C_m$$ . In this paper, we show that $$R(T_n,W_{s,4})=(n-1)(s+1)+1$$ for $$n\ge 3,s\ge 2$$ and $$R(T_n,W_{s,5})=(n-1)(s+2)+1$$ for $$n\ge 3,s\ge 1$$ . These generalize some known results on Ramsey numbers for a tree versus a wheel.

In the era of big data, with the increase of data processing information and the increase of data complexity, higher requirements are put on the tools and algorithms of data processing. As ...

For a graph G = ( V , E ) , the eccentric connectivity index of G, denoted by ξc(G), is defined as ξ c ( G ) = ∑ v ∈ V ɛ ( v ) d ( v ) , where ɛ(v) and d(v) are the eccentricity and the degree of v in G, respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G, and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G, and characterize all extremal graphs.

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph G is named as an all fractional (g, f, n′, m)-critical deleted graph if the remaining subgraph keeps being an all fractional (g, f, m)-critical graph, despite experiencing the removal of arbitrary n′ vertices of G. In this paper, we study the relationship between neighborhood conditions and a graph to be all fractional (g, f, n′, m)-critical deleted. Two sufficient neighborhood conditions are determined, and furthermore we show that the conditions stated in the main results are sharp.

Recently, many exciting achievements have been made in improving the cycling performance of SnO2-based electrode. However, the low initial Coulombic efficiency of SnO2-based electrode caused by the formation of solid electrolyte interphase and some of the irreversible formation of Li2O in SnO2 structure still remains unresolved. Here this work reports a primary-cell process to prelithiate the SnO2-based composites to compensate the irreversible loss during the first cycle. The primary-cell-prelithiated SnO2@C composites performs a high initial Coulombic efficiency from 92.7% to >100% with the prelithiation time extends. Even after 600 cycles, the prelithiated (24 h) SnO2@C composites show a high capacity retention of 95% (848 mAh g−1) at a current density of 0.5 A g−1.