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In recent years, drought disaster has occurred frequently in China, causing significant agricultural losses. It is increasingly important to assess the risk of agricultural drought disaster (ADD) and to develop a targeted risk management approach. In this study, an ADD risk assessment model was established. First, an improved fuzzy analytic hierarchy process based on an accelerated genetic algorithm (AGA-FAHP) was used to build an evaluation indicator system. Then, based on the indicators, the ADD assessment connection numbers were established using the improved connection number method. Finally, the entropy information diffusion method was used to form an ADD risk assessment model. The model was applied to the Huaibei Plain in Anhui Province (China), with the assessment showing that, in the period from 2008 to 2017, the plain was threatened continuously by ADD, especially during 2011&ndash, 2013. The risk assessment showed that southern cities of the study area were nearly twice as likely to be struck by ADD as northern cities. Meanwhile, the eastern region had a higher frequency of severe and above-grade ADD events (once every 21 years) than the western region (once every 25.3 years). Therefore, Huainan was identified as a high-risk city and Huaibei as a low-risk city, with Suzhou and Bengbu more vulnerable to ADD than Fuyang and Bozhou. Understanding the spatial dynamics of risk in the study area can improve agricultural system resilience by optimizing planting structures and by enhancing irrigation water efficiency. This model could be used to provide support for increasing agricultural drought disaster resilience and risk management efficiency.

As a sophisticated computing unit, the pyramidal neuron requires sufficient metabolic energy to fuel its powerful computational capabilities. However, the majority of previous works focus on nonlinear integration and energy consumption in individual pyramidal neurons but seldom on the effects of metabolic energy on synaptic transmission and dendritic integration. Here, we developed biologically plausible models to simulate the synaptic transmission and dendritic integration of pyramidal neurons, exploring the relations between synaptic transmission and metabolic energy and between dendritic integration and metabolic energy. We find that synaptic energy not only drives synaptic vesicle cycle, but also participates in the regulation of this cycle. Release probability of synapses adapts to synaptic energy levels by regulating the speed of synaptic vesicle cycle. Besides, we also find that to match neural energy levels, only a part of the synapses receive presynaptic signals during a given period so that neurons have a low action potential frequency. That is, the number of simultaneously active synapses over a period of time should be adapted to neural energy levels.

Among the many interesting problems of determining the proper connection numbers of graphs, it is worthwhile to study the proper connection number of G according to some constraints of the complementary graph.

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $k\geq3$. If $|E(G)|\geq \binom{n-m-(k+1-m)(\delta+1)}{2} +(k+1-m)\binom{\delta+1}{2}+k+2$, then $pc(G)\leq k$, where $m$ takes the value $t$ if $\delta=1$ and $\lfloor \frac{k}{\delta-1} \rfloor$ if $\delta\geq2$. Furthermore, if $k=2$ and $\delta=2$, %(i.e., $|E(G)|\geq \binom{n-5}{2} +7$) $pc(G)\leq 2$, except $G\in \{G_{1}, G_{n}\}$ ($n\geq8$), where $G_{1}=K_{1}\vee 3K_{2}$ and $G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_{1}\vee (2K_{2}$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}\vee (2K_{2}$) being joined. If $k=2$, $\delta \geq 3$, we conjecture $pc(G)\leq 2$, where $m$ takes the value $1$ if $\delta=3$ and $0$ if $\delta\geq4$ in the assumption., Comment: 12. arXiv admin note: text overlap with arXiv:1601.04162 and arXiv:1602.07163 by other authors

An edge-colored connected graph G is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors needed to color the edges of G to make it properly connected. In this work, we obtain sharp lower bounds for \(pc(G)+ pc(\overline {G})\), and \(pc(G)pc(\overline {G})\), where G is a connected graph of order at least 8. Among our results, we also get sharp lower bounds for \(pvc(G)+pvc(\overline {G})\), \(pvc(G)pvc(\overline {G})\), \(ptc(G)+ptc(\overline {G})\) and \(ptc(G)ptc(\overline {G})\), where pvc(G) and ptc(G) are proper vertex-connection number and proper total-connection number of G, respectively.

A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored with one same color. An edge-coloring of G is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in G. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors used in an MC-coloring of G. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we study two kinds of Erdźs---Gallai-type problems for mc(G), and completely solve them.

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.

Marshalling station is extremely vital to the railway transportation, marshalling stations will inevitably be attacked in military actions, so the damage effect of marshalling station was researched here. The sub-targets were analyzed and the damage model was established based on the characteristics of railway marshalling station, the damage circle was used to calculate the damage probability. On the basis of this process, the evaluation model of network diagram was constructed, the reasonable integration of cumulative, cascade and multiple functions of damage effect was realized. The damage effect was calculated by using the connection number method, and the example was analyzed. The result can give an accurate evaluation on the damage about system function of the marshalling station target and give a reasonable foundation for command decision-making.

An edge colored graph G = (V(G), E(G)) is said rainbow connected, if any two vertices are connnected by a path whose edges have distinct colors. The rainbow connection number of G, denoted by rc(G), is the smallest positive integer of colors needed in order to make G rainbow connected. The vertex-colored graph G is said rainbow vertex-connected, if for every two vertices u and v in V(G), there is a u-v path with all internal vertices have distinct color. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors needed in order to make G rainbow vertex-connected. In this paper, we determine rainbow (vertex) connection number of pencil graphs.

The (directed) proper connection number of a given (di)graph G is the least number of colors needed to edge-color G such that there exists a properly colored (di)path between every two vertices in G . There also exist vertex-coloring versions of the proper connection number in (di)graphs. We initiate the study of the complexity of computing the proper connection number and (two variants of) the proper vertex connection number, in undirected and directed graphs, respectively. First we disprove some conjectures of Magnant et al. (2016) on characterizing strong digraphs with directed proper connection number at most two. In particular, we prove that deciding whether a given digraph has directed proper connection number at most two is NP-complete. Furthermore, we show that there are infinitely many such digraphs without an even-length dicycle. To the best of our knowledge, the proper vertex connection number of digraphs has not been studied before. We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, on a more positive side we present polynomial-time recognition algorithms for bounded-treewidth graphs and bipartite graphs with proper connection number at most two.

A path in an edge-colored graph is called conflict-free if it contains a color that is used by exactly one of its edges. An edge-colored graph G is conflict-free connected if for any two distinct vertices of G, there is a conflict-free path connecting them. For a connected graph G, the conflict-free connection number of G, denoted by cfc(G), is defined as the minimum number of colors that are required to make G conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We use L(G) to denote the line graph of a graph G. In general, the k-iterated line graph of a graph G, denoted by \(L^k(G)\), is the line graph of the graph \(L^{k-1}(G)\), where \(k\ge 2\) is a positive integer. We first show that for an arbitrary connected graph G, there exists a positive integer k such that \(cfc(L^k(G))\le 2\). Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph G and an arbitrary positive integer k, we always have \(cfc(L^{k+1}(G))\le cfc(L^k(G))\), with only the exception that G is isomorphic to a star of order at least 5 and \(k=1\). Finally, we obtain the exact values of \(cfc(L^k(G))\), and use them as an efficient tool to get the smallest nonnegative integer \(k_0\) such that \(cfc(L^{k_0}(G))=2\).

Let k be a positive integer and G be a k -connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k -connection number r c k ( G ) of G . An edge-coloured path is rainbow if its edges have distinct colours. Then, r c k ( G ) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function r c k ( G ) has since been studied by numerous researchers. An analogue of the function r c k ( G ) involving vertex colourings, the rainbow vertex k -connection number r v c k ( G ) , was subsequently introduced. In this paper, we introduce a version which involves total colourings. A total-coloured path is total-rainbow if its edges and internal vertices have distinct colours. The total rainbow k -connection number of G , denoted by t r c k ( G ) , is the minimum number of colours required to colour the edges and vertices of G , so that any two vertices of G are connected by k internally vertex-disjoint total-rainbow paths. We study the function t r c k ( G ) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions r c k ( G ) , r v c k ( G ) , and t r c k ( G ) , by considering how close and how far apart t r c k ( G ) can be from r c k ( G ) and r v c k ( G ) .

The rainbow vertex-connection number, r v c ( G ) , of a connected graph G is the minimum number of colors needed to color its vertices such that every pair of vertices is connected by at least one path whose internal vertices have distinct colors. In this paper we prove that for a 2-connected graph G of order n , r v c ( G ) ≤ { ⌈ n / 2 ⌉ − 2 if n = 3 , 5 , 9 ⌈ n / 2 ⌉ − 1 if n = 4 , 6 , 7 , 8 , 10 , 11 , 12 , 13 or 15 ⌈ n / 2 ⌉ if n ≥ 16 or n = 14 . The upper bound is tight since the cycle C n on n vertices has its r v c ( C n ) equal to this bound.

A model of risk trend assessment for metro tunnel crossing underground pipelines is proposed in this paper based on the partial connection number. This paper fully considered the certainty and uncertainty of the index and used "partial connection number method" to analyze the risk developing trend and its sort of metro tunnel crossing underground pipelines. A case study was performed on a metro tunnel engineering of Shenzhen Line 9. The results showed that this model considered the dynamic characteristics of risk, enriching the contents of risk assessment of underground engineering, which can provide a reference for the risk management.

Assessment and diagnosis of regional agricultural drought resilience (RADR) is an important groundwork to identify the shortcomings of regional agriculture to resist drought disasters accurately. In order to quantitatively assess the capacity of regional agriculture system to reduce losses from drought disasters under complex conditions and to identify vulnerability indexes, an assessment and diagnosis model for RADR was established. Firstly, this model used the improved fuzzy analytic hierarchy process to determine the index weights, then proposed an assessment method based on connection number and an improved connection entropy. Furthermore, the set pair potential based on subtraction was used to diagnose the vulnerability indexes. In addition, a practical application had been carried out in the region of the Huaibei Plain in Anhui Province. The evaluation results showed that the RADR in this area from 2005 to 2014 as a whole was in a relatively weak situation. However, the average grade values had decreased from 3.144 to 2.790 during these 10 years and the RADR had an enhanced tendency. Moreover, the possibility of RADR enhancement for six cities in this region decreased from east to west, and the drought emergency condition was the weak link of the RADR in the Huaibei Plain.

The safety management system assessment (SMSA) model was built up on the basis of set-pair analysis (SPA) theory and its effectiveness was verified by practical application research. The assessment model of five-element identical-discrepant-antithetical connection number vector was formed in that the indicators were divided into five grades. The calculation of connection number component was simplified as rigorous rules were made for the grades division of indicators. The judging rule of grading safety management system (SMS) was established under the principal of confidence level. The applied research was conducted, which demonstrated that the SPA based assessment model could provide credible conclusion because the results highly agreed with those of BNWSS audition. In addition, this model can be applied to assess SMS of all levels and makes scientific judgment for future state of SMS.

Local protectionism is a barrier to the sound development of the market economy. To analysis the impact and effectiveness of local protectionism from the angle of mathematic or system science, this paper analysis the connection function of local economy system based on set pair analysis method and the basic ideas of Jieke Theory. Then, the negative impact of local protectionism was discussed according to the connection function between regional economies. Finally, some suggestions were given.

The research of water ecological security evaluation has important practical significance. This paper integrates the multivariate connection number with Markov chain to construct a dynamic evaluation model and set up the water ecological security evaluation index system from four aspects: water resources, water ecological environment, water management and control, social economy. The proposed model and evaluation index system are applied to respectively evaluate the water ecological security of Shandong Province in 1993, 1999, 2005 and 2011, and forecast its change trend, the results indicate that the proposed model has a efficient performance, the "relatively insecure" grade and "relatively secure" grade alternated in the past, the steady-state level of the water ecological security grade in Shandong province tends to be "relatively insecure", and the trend of water ecological system would rise to "relatively insecure" in the future.

The evaluation results of power system are greatly influenced by the reliability parameters and uncertainty of system components. The connection number assessment model and an approach have been presented to assess the occurrence frequency due to voltage sags. The proposed method had been applied to a real distribution system. Compared with the interval number method, the simulation results have shown that this method is simple and flexible.

A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard. We also show that for any fixed integer $k\geq 2$, this problem belongs to NP-class, and so it becomes NP-Complete., Comment: 10 pages

A path in an edge-colored graph is called rainbow if no two edges of it are colored the same. For an l -connected graph G and an integer k with 1 ≤ k ≤ l , the rainbow k -connection number r c k ( G ) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. Fujita et al. proposed a problem: What is the minimum constant α > 0 such that for every 2-connected graph G on n vertices, we have r c 2 ( G ) ≤ α n ? In this paper, we prove that the minimum constant α = 1 and r c 2 ( G ) = n if and only if G is a cycle of order n , which solves the problem of Fujita et al.

An edge coloring of a connected graph G is a proper-path coloring if every two vertices of G are connected by a properly colored path. The minimum number of colors required of a proper-path coloring of G is called the proper connection number pc(G) of G. For a connected graph G with proper connection number 2, the minimum size of a connected spanning subgraph H of G with pc(H) = 2 is denoted by μ(G). It is shown that if s and t are integers such that t ≥ s + 2 ≥ 5, then μ(K_{s,t} ) = 2t − 2. We also determine μ(G) for several classes of complete multipartite graphs G. In particular, it is shown that if G = K_{n_1, n_2, ..., n_k} is a complete k-partite graph, where k ≥ 3, r = \sum^{k−1}_{i=1} n_i ≥ 3 and t = n_k ≥ r^2 + r, then μ(G) = 2t − 2r + 2.

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same color. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph. For a connected graph $G$, the {\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$. These concepts are inspired by the concepts of monochromatic connection number $mc(G)$, monochromatic vertex connection number $mvc(G)$ and total rainbow connection number $trc(G)$ of a connected graph $G$. Let $l(T)$ denote the number of leaves of a tree $T$, and let $l(G)=\max\{ l(T) | $ $T$ is a spanning tree of $G$ $\}$ for a connected graph $G$. In this paper, we show that there are many graphs $G$ such that $tmc(G)=m-n+2+l(G)$, and moreover, we prove that for almost all graphs $G$, $tmc(G)=m-n+2+l(G)$ holds. Furthermore, we compare $tmc(G)$ with $mvc(G)$ and $mc(G)$, respectively, and obtain that there exist graphs $G$ such that $tmc(G)$ is not less than $mvc(G)$ and vice versa, and that $tmc(G)=mc(G)+l(G)$ holds for almost all graphs. Finally, we prove that $tmc(G)\leq mc(G)+mvc(G)$, and the equality holds if and only if $G$ is a complete graph., Comment: 12 pages

Because fruit state need to be discriminate to select the different harvesting method before it is fast picked, the fruit state discrimination method was developed. Firstly, the two apple images were acquired, and then segmented by the improved OTSU dynamic threshold segmentation method. Secondly, the picking target fruit was selected based on the principle of the nearest distance to image center when there were a lot of fruits. Thirdly, the two segmented fruit images were done by the inter-frame difference method, then the target fruit state was got using the connection number discrimination to the difference image and the centroid coordinates matching discrimination to the oscillating fruit image. Lastly, the test results showed that the designed algorithm was feasible and effective for most cases in natural environment, and the discrimination time was less than 0.2s.

PurposeSince the present university leading cadre capability evaluations are mainly static, there is a lack of evaluation methods to address dynamic and multi‐scale‐based situations. Some important factors to address people's performances, such as the developing trends and potentialities are considered. The purpose of this paper is to attempt to propose a dynamic evaluation model based on multivariate delay connection number.Design/methodology/approachFirst, evaluation results of previous periods are taken into the current period and values of uncertain numbers are revised. Next, a dynamic evaluation model is built for university leading cadre. Last, a numerical example is used to prove its feasibility and effectiveness.FindingsThis paper promotes a corrected equal portion value method of uncertain numbers i1(τ), … ,in(τ) from the adjacent two periods to multi periods, and builds a dynamic evaluation model, which proved simple‐structured and easy to apply.Practical implicationsThe new model is applied in the evaluation of university leading cadre, and could give a more comprehensive, dynamic, scientific analysis for evaluation by model application.Originality/valueThe paper develops a novel evaluation model based on a multivariate delay connection number, which expands traditional methods from a static situation to a dynamic and multi‐scale‐based situation.

By applying neural networks to forecasting Beijing motor vehicles sold, sequencing the principal factors and analyzing the development trend using connection number and partial connection number of the set pair analysis (SPA), we set up the forecasting model of Beijing motor vehicles sold. The instance analysis shows that it is a scientific and suitable system analyzing method of high forecasting accuracy.

Risk analysis of swell and shrinkage is essential to properly evaluate engineering stability and safety of expansive soils. Herein a novel model for risk analysis of swell and shrinkage of expansive soils was proposed by means of coupling set pair analysis with stochastic simulation of triangular fuzzy numbers. A concept of multi-element connection number for a set pair, which consisted of measured evaluation index sets and classification standard sets of shrink and expansion grade, was described to express hierarchy and fuzziness of membership between the evaluation sample and the classification standard. Moreover, triangular fuzzy numbers simulated by the stochastic simulation method were presented to depict the changing process and fuzziness of component coefficients of discrepancy degree. Combined with the weight of the evaluation index, integrated connection number was calculated to specify the swell and shrinkage grade of the evaluation sample. A confidence level of evaluation result was also given corresponding to a confidence interval. Finally, a practical example was used to confirm and to compare with the fuzzy mathematic method and extension method. The results show that this presented method is more feasible and easy to operate, and the result is good. It can provide a reliable degree of evaluation results for risk design and risk management of projects on expansive soils.

In view of the problem that seismic damage evaluation for reinforced concrete structure are difficult to quantify, a new model for seismic damage evaluation was established by five-element connection number based on set pair analysis in the paper, which took internal damage index, maximum inter-storey drift, number of inelastic cycles and energy-concentrated quotient as evaluation factors. All indexes can be summarized into five-element connection number. Four different types of structure are analyzed on the model with more satisfactory results. Compared with the matter-element analysis evaluation method, this model is perfect, the evaluating result is more reasonable and its resolving power is higher. By this practical example, the model proved to be effective in seismic damage evaluation on reinforced concrete structure.

Set pair analysis (SPA) is emerging as a new system theory method for analyzing vagueness and uncertainty of facts. Many problems of uncertainty of fuzziness can be solved with the application of identity-discrepancy-contrary expression, and five-element connection number (FECN) is an extension of SPA. Aiming at the problem being too complex for the present evaluation methods on lake eutrophication, an assessment of lake eutrophication based on situation of five-element connection number was suggested. The results of the assessment are basically in concordance with other evaluation methods. So, the evaluation model provides realistic support for remediation-related decisions for water resource management.

The optimization of insulation material of external wall is a multi-attribute uncertainty decision-making problem. Herein based on theory of multi-attribute decision making, a novel decision-making model using generalized connection number of set pair was discussed to solve the problem. Namely, the certainty and uncertainty relationship between the evaluation plan and the optimal plan or the worst plan for the optimization of insulation material was described by means of identical-contrary analysis of set pair theory. And a generalized connection degree was introduced to evaluate comprehensively the relative membership degree of the evaluation plan to the optimal plan. The results from a practical example show that the proposed model used to select the plan of insulation material is feasible and reliable, and easy to operate.

A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

Linguistic intuitionistic fuzzy set (LIFS) is the better way to deal with the uncertain and imprecise information in group decision-making problems. On the other hand, the set pair analysis (SPA) theory provides a quantitative analysis to integrate the certainty and uncertainties as a combined system by defining the connection number corresponding to it. In the present paper, we have enhanced the LIFS with the SPA theory and hence defined the linguistic connection number (LCN) and its various operational laws. Based on it, we have developed various aggregation operators, namely LCN weighted geometric, LCN ordered weighted geometric, and LCN hybrid geometric operators with LIFS environment. Also, the shortcoming of the existing operators under LIFS environment has been highlighted and overcomes by the proposed operators. Few properties of these operators have been also investigated. Further, a group decision-making approach has been presented, based on these operators, which has been illustrated by a numerical example to show the effectiveness and validity of the proposed approach.

Spike synchrony of the neural system is thought to have very dichotomous roles. On the one hand, it is ubiquitously present in the healthy brain and is thought to underlie feature binding during information processing. On the other hand, large scale synchronization is an underlying mechanism of epileptic seizures. In this paper, we investigate the spike synchrony of Hindmarsh–Rose (HR) neural networks. Our focus is the influence of the network connections on the spike synchrony of the neural networks. The simulations show that desynchronization in the nearest-neighbor coupled network evolves into accurate synchronization with connection-rewiring probability p increasing. We uncover a phenomenon of enhancement of spike synchrony by randomly rewiring connections. With connection strength c and average connection number m increasing spike synchrony is enhanced but it is not the whole story. Furthermore, the possible mechanism behind such synchronization is also addressed.

Because connection number can express and process synthetic uncertainties caused by various uncertainties in the transmission network planning, a connection number model (CNM) was presented to compare the values of connection number logically. This paper proposed a novel model for transmission network flexible planning with uncertainty. In the proposed planning model both certainty and uncertainty information were included, and the cost-benefit analysis method was used to evaluate the candidate schemes in the objective function. Its good adaptability and flexibility were illustrated through two examples.

We investigate the dynamics of network minority games on Kauffman's NK networks (Kauffman nets) and growing directed networks (GDNets). We show that the dynamics and the associated phase structure of the game depend crucially on the structure of the underlying network. The dynamics on GDNets is very stable for all values of the connection number K, in contrast to the dynamics on Kauffman's NK networks, which becomes chaotic when K > Kc= 2. For Kauffman nets with K > 3, the evolutionary scheme has no effect on the dynamics (it remains chaotic) and the performance of the MG resembles that of a random choice game (RCG).

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. In this paper, we show that almost all graphs have the proper connection number 2. More precisely, let $G(n,p)$ denote the Erd\"{o}s-R\'{e}nyi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independent from other pairs. We prove that for sufficiently large $n$, $pc(G(n,p))\le2$ if $p\ge\frac{\log n +\alpha(n)}{n}$, where $\alpha(n)\rightarrow \infty$., Comment: 13 pages

The first (7,8)-connected polyoxometalate-based metal–organic framework (POM–MOF) has been constructed from seven- and eight-connected twin Wells–Dawson clusters, and possesses the highest connection number of polyoxometalates to any mixed-connected POM–MOF to date and a unique structural motif that contains both organic–inorganic and all-inorganic networks.

The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree} if the vertices of $V(T)\setminus S$ have distinct colors. For a fixed integer $k$ with $2\leq k\leq n$, the vertex-coloring $c$ of $G$ is called a \emph{$k$-vertex-rainbow coloring} if for every $k$-subset $S$ of $V(G)$ there exists a vertex-rainbow $S$-tree. In this case, $G$ is called \emph{vertex-rainbow $k$-tree-connected}. The minimum number of colors that are needed in a $k$-vertex-rainbow coloring of $G$ is called the \emph{$k$-vertex-rainbow index} of $G$, denoted by $rvx_k(G)$. When $k=2$, $rvx_2(G)$ is nothing new but the vertex-rainbow connection number $rvc(G)$ of $G$. In this paper, sharp upper and lower bounds of $srvx_k(G)$ are given for a connected graph $G$ of order $n$,\ that is, $0\leq srvx_k(G)\leq n-2$. We obtain the Nordhaus-Guddum results for $3$-vertex-rainbow index, and show that $rvx_3(G)+rvx_3(\overline{G})=4$ for $n=4$ and $2\leq rvx_3(G)+rvx_3(\overline{G})\leq n-1$ for $n\geq 5$. Let $t(n,k,\ell)$ denote the minimal size of a connected graph $G$ of order $n$ with $rvx_k(G)\leq \ell$, where $2\leq \ell\leq n-2$ and $2\leq k\leq n$. The upper and lower bounds for $t(n,k,\ell)$ are also obtained., Comment: 12 pages, 4 figures

BitTorrent (BT) is a widely used P2P file-sharing system, and its performance is a hot research topic. It is important to design an efficient BT topology measurement system for BT performance research. Some studies use the BT clients to collect the information of BT topology, which may be incomplete due to their limitation. In this paper, combining the passive and active approaches, we design a BT topology measurement system. We evaluate the peer coverage ratio by comparing the number of nodes achieved by the measurement system with that registered to the tracker. And we evaluate the connection coverage ratio by comparing the connection number extracted from PEX messages with that of BT clients. The results show that the average peer coverage ratio can reach above 95%, which is much higher than that of μTorrent, and the connection coverage ratio reaches above 90%. Overall, the performance of our measurement system is better than that of BT clients.

Generation system reliability assessment not only has its macro-certain factor, but also has its uncertain factor. Uncertainty factor gets reliability parameters fluctuating around its expectation. The connection mathematics, developed from the set pair analysis, can well deal with the macro-certain factor and the uncertain factor in power system generation reliability assessment. The connection number is applied for generation system reliability assessment in this paper. The identical-discrepancy connection number (IDCN) is used to represent reliability parameters. Quantitative method of reliability evaluation, based on the state space method, is used to evaluate generation adequacy of power system. IEEE-RTS79 generation system is assessed and the result is discussed. The method suggested in this paper gives a new idea to deal with the uncertainty of parameters in reliability evaluation of power system.

All the above parameters on rainbow connections involved edge-colorings of graphs. A natural idea is to introduce a similar parameter involving vertex-colorings of graphs. It is, as mentioned above, a vertex version of the rainbow connection number. Krivelevich and Yuster (J. Graph Theory 63(3):185–191, 2009) is the first to propose this new concept and proved a theorem analogous to the result of Theorem 3.1.10.

Scientific evaluation to the quality of teaching is an indispensable part in the process of teaching management. But the measure of teacher’s classroom teaching is fuzzy and difficult to be clearly decided. The index system on evaluating the classroom teaching quality is given; a new classroom teaching quality evaluation method is presented based on multi-element connection number by using the theory of set pair analysis. It overcomes shortages of original classroom teaching quantity evaluation work, such as subjective and arbitrary and offers scientific, objective, fair, accurate evaluations. A new method and thought is provided for classroom teaching quantity evaluation work.

Based on the connection number in the Set Pair Analysis, this article proposes a network model of airport battle damage assessment (BDA). Comprehensive analysis of the advantages and disadvantages of the previous algorithms on BDA is first implemented. In order to deal with the uncertainties of BDA, the connection number in the Set Pair Analysis is introduced to calculate battle damage, and the partial connection number is applied to predict the development of BDA. The network model based on the promoting or hindering causal relations in this paper is improved to be more effective.