Perfect 1-k Matchings of Bipartite Graphs

Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M. A perfect 1-k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M, and a vertex in X can be incident with an arbitrary number of edges in M. In this paper, we give three sufficient and necessary conditions for the existence of perfect 1-k matchings and for the existence of 1-k matchings covering | X |−dvertices in X, respectively, and characterize k-elementary bipartite graph which is a graph such that the subgraph induced by all k-allowed edges is connected, where an edge is k-allowed if it is contained in a perfect 1-k matching.

PARTITION A GRAPH WITH SMALL DIAMETER INTO TWO INDUCED MATCHINGS

Given a simple graph G and a positive integer k, the induced matching k-partition problem asks whether there exists a k-partition (V 1, V 2, ..., V k) of V(G) such that for each i(1≤i≤k), G[V i] is 1-regular. This paper studies the computational complexity of this problem for graphs with small diameters. The main results are as follows: Induced matching 2-partition problem of graphs with diameter 6 and induced matching 3-partition problem of graphs with diameter 2 are NP-complete; induced matching 2-partition problem of graphs with diameter 2 is polynomially solvable.

Perfect matchings on a type of lattices with toroidal boundary

Enumeration of perfect matchings on graphs has a longstanding interest in combinatorial mathematics. In this paper, we obtain some explicit expressions of the number of perfect matchings for a type of Archimedean lattices with toroidal boundary by applying Tesler's crossing orientations to obtain some Pfaffan orientations and enumerating their Pfaffans.

ON MAXIMAL MATCHINGS OF CONNECTED GRAPHS

Let I with |I| = k be a matching of a graph G (briefly, I is called a k-matching). If I is not a proper subset of any other matching of G, then I is a maximal k-matching and m(gk, G) is used to denote the number of maximal k-matchings of G. Let gk be a k-matching of G, if there exists a subset {e1, e2,…, ei} of E(G) \ gk, i (?)1, such that (1) for any j ∈ {1, 2,…,i}, gk + {ej} is a (k + l)-matching of G; (2) for any f ∈ E(G) \ (gk ∪ {e1,e2,…,ei}), gk + {f} is not a matching of G; then gk, is called an i wings k-matching of G and mi(gk,G) is used to denote the number of i wings k-matchings of G. In this paper, it is proved that both mi(gk,G) and m(gk,G) are edge reconstructible for every connected graph G, and as a corollary, it is shown that the matching polynomial is edge reconstructible.

Cover a 3-regular Claw-free Graph by Induced Matchings

The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪···∪V(Mk)covers V(G).This paper shows that,if G is a 3-regular claw-free graph,then imc(G)∈{2,3}.

Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.

An o(n^2.5) Algorithm: For Maximum Matchings in General Graphs

This article extend the John E. Hopcroft and Richart M. Karp Algorithm (HK Algorithm) for maximum matchings in bipartite graphs to the non-bipartite case by providing a new approach to deal with the blossom in alternating paths in the process of searching for augmenting paths, which different from well-known “shrinking” way of Edmonds and makes the algorithm for maximum matchings in general graphs more simple.

Rainbow Matchings in Properly Colored Bipartite Graphs

Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition (X,Y) and . We show that if , then G has a rainbow coloring of size at least .

The Number of Perfect Matchings in (3,6)-Fullerene

A(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons,and has the connectivity 2 or 3.The(3,6)-fullerenes with connectivity 2 are the tubes consisting of l concentric hexagonal layers such that each layer consists of two hexangons,capped on each end by two adjacent triangles,denoted by T_(l)(l≥1).A(3,6)-fullerene Tl with n vertices has exactly 2n/4+1 perfect matchings.The structure of a(3,6)-fullerene G with connectivity 3 can be determined by only three parameters r,s and t,thus we denote it by G=(r,s,t),where r is the radius(number of rings),s is the size(number of spokes in each layer,s(≥4,s is even),and t is the torsion(0≤t<s,t≡r mod 2).In this paper,the counting formula of the perfect matchings in G=n+1,4,t)is given,and the number of perfect matchpings is obtained.Therefore,the correctness of the conclusion that every bridgeless cubic graph with p vertices has at least 2p/3656perfect matchings proposed by Esperet et al is verified for(3,6)-fullerene G=(n+1,4,t).

Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

Let G be a simple graph with 2n vertices and a perfect matching.The forcing number f(G,M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G.Among all perfect matchings M of G,the minimum and maximum values of f(G,M) are called the minimum and maximum forcing numbers of G,denoted by f(G) and F(G),respectively.Then f(G)≤F(G) ≤n-1.Che and Chen(2011) proposed an open problem:how to characterize the graphs G with f(G)=n-1.Later they showed that for a bipartite graph G,f(G)=n-1 if and only if G is complete bipartite graph K_(n,n).In this paper,we completely solve the problem of Che and Chen,and show that f(G)=n-1 if and only if G is a complete multipartite graph or a graph obtained from complete bipartite graph K_(n,n) by adding arbitrary edges in one partite set.For all graphs G with F(G)=n-1,we prove that the forcing spectrum of each such graph G forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs G form an integer interval from [n/2] to n-1.

Interface Engineering of Titanium Nitride Nanotube Composites for Excellent Microwave Absorption at Elevated Temperature

Currently,the microwave absorbers usually suffer dreadful electromagnetic wave absorption(EMWA)performance damping at elevated temperature due to impedance mismatching induced by increased conduction loss.Consequently,the development of high-performance EMWA materials with good impedance matching and strong loss ability in wide temperature spectrum has emerged as a top priority.Herein,due to the high melting point,good electrical conductivity,excellent environmental stability,EM coupling effect,and abundant interfaces of titanium nitride(TiN)nanotubes,they were designed based on the controlling kinetic diffusion procedure and Ostwald ripening process.Benefiting from boosted heterogeneous interfaces between TiN nanotubes and polydimethylsiloxane(PDMS),enhanced polarization loss relaxations were created,which could not only improve the depletion efficiency of EMWA,but also contribute to the optimized impedance matching at elevated temperature.Therefore,the TiN nanotubes/PDMS composite showed excellent EMWA performances at varied temperature(298-573 K),while achieved an effective absorption bandwidth(EAB)value of 3.23 GHz and a minimum reflection loss(RLmin)value of−44.15 dB at 423 K.This study not only clarifies the relationship between dielectric loss capacity(conduction loss and polarization loss)and temperature,but also breaks new ground for EM absorbers in wide temperature spectrum based on interface engineering.

An improved particle filter indoor fusion positioning approach based on Wi-Fi/PDR/geomagnetic field

The existing indoor fusion positioning methods based on Pedestrian Dead Reckoning(PDR)and geomagnetic technology have the problems of large initial position error,low sensor accuracy,and geomagnetic mismatch.In this study,a novel indoor fusion positioning approach based on the improved particle filter algorithm by geomagnetic iterative matching is proposed,where Wi-Fi,PDR,and geomagnetic signals are integrated to improve indoor positioning performances.One important contribution is that geomagnetic iterative matching is firstly proposed based on the particle filter algorithm.During the positioning process,an iterative window and a constraint window are introduced to limit the particle generation range and the geomagnetic matching range respectively.The position is corrected several times based on geomagnetic iterative matching in the location correction stage when the pedestrian movement is detected,which made up for the shortage of only one time of geomagnetic correction in the existing particle filter algorithm.In addition,this study also proposes a real-time step detection algorithm based on multi-threshold constraints to judge whether pedestrians are moving,which satisfies the real-time requirement of our fusion positioning approach.Through experimental verification,the average positioning accuracy of the proposed approach reaches 1.59 m,which improves 33.2%compared with the existing particle filter fusion positioning algorithms.

Understanding the Relationship Between Shrinking Cities and Land Prices:Spatial Pattern,Effectiveness,and Policy Implications

Urban shrinkage has emerged as a widespread phenomenon globally and has a significant impact on land,particularly in terms of land use and price.This study focuses on 2851 county-level cities in China in 2005–2018(excluding Hong Kong,Macao,Taiwan,and‘no data’areas in Qinhai-Tibet Plateau)as the fundamental units of analysis.By employing nighttime light(NTL)data to identify shrinking cities,the propensity score matching(PSM)model was used to quantitatively examine the impact of shrinking cities on land prices,and evaluate the magnitude of this influence.The findings demonstrate the following:1)there were 613 shrinking cities in China,with moderate shrinkage being the most prevalent and severe shrinkage being the least.2)Regional disparities are evident in the spatial distribution of shrinking cities,especially in areas with diverse terrain.3)The spatial pattern of land price exhibits a significant correlated to the economic and administrative levels.4)Shrinking cities significantly negatively impact on the overall land price(ATT=–0.1241,P<0.05).However,the extent of the effect varies significantly among different spatial regions.This study contributes novel insights into the investigation of land prices and shrinking cities,ultimately serving as a foundation for government efforts to promote the sustainable development of urban areas.

Preparation of CIP@TiO_(2) composite with broadband electromagnetic wave absorption properties

Scholars aim for the improved impedance matching (Z) of materials while maintaining their excellent wave absorption properties. Based on the hydrolysis characteristics of isopropyl titanate, a simple preparation process for the coating of carbonyl iron powder(CIP) with TiO_(2) was designed. Given the TiO2coating, the Z of the CIP@TiO_(2) composite was adjusted well by decreasing the dielectric constant. Moreover, the interfacial polarization of CIP@TiO_(2) was enhanced. Ultimately, the electromagnetic-wave (EMW) absorption property of the CIP@TiO_(2)composite was improved substantially, the minimum reflection loss reached-46.07 dB, and the effective absorption bandwidth can reach 8 GHz at the composite thickness of 1.5 mm. Moreover, compared with CIP, the oxidation resistance of CIP@TiO_(2)showed remarkable improvement. The results revealed that the oxidation starting temperature of CIP@TiO_(2) as about 400℃,whereas the uncoated CIP had an oxidation starting temperature of approximately 250℃. Moreover, the largest oxidation rate temperature of CIP@TiO_(2) increased to around 550℃. This work opens up a novel strategy for the production of high-performance EMW absorbers via structural design.

From VIB‑to VB‑Group Transition Metal Disulfides:Structure Engineering Modulation for Superior Electromagnetic Wave Absorption

The laminated transition metal disulfides(TMDs),which are well known as typical two-dimensional(2D)semiconductive materials,possess a unique layered structure,leading to their wide-spread applications in various fields,such as catalysis,energy storage,sensing,etc.In recent years,a lot of research work on TMDs based functional materials in the fields of electromagnetic wave absorption(EMA)has been carried out.Therefore,it is of great significance to elaborate the influence of TMDs on EMA in time to speed up the application.In this review,recent advances in the development of electromagnetic wave(EMW)absorbers based on TMDs,ranging from the VIB group to the VB group are summarized.Their compositions,microstructures,electronic properties,and synthesis methods are presented in detail.Particularly,the modulation of structure engineering from the aspects of heterostructures,defects,morphologies and phases are systematically summarized,focusing on optimizing impedance matching and increasing dielectric and magnetic losses in the EMA materials with tunable EMW absorption performance.Milestones as well as the challenges are also identified to guide the design of new TMDs based dielectric EMA materials with high performance.

Shared genetic architecture highlights the bidirectional association between major depressive disorder and fracture risk

Background There is limited evidence suggesting that osteoporosis might exacerbate depressive symptoms,while more studies demonstrate that depression negatively affects bone density and increases fracture risk.Aims To explore the relationship between major depressive disorder(MDD)and fracture risk.Methods We conducted a nested case-control analysis(32670 patients with fracture and 397017 individuals without fracture)and a matched cohort analysis(16496 patients with MDD and 435492 individuals without MDD)in the same prospective UK Biobank data set.Further,we investigated the shared genetic architecture between MDD and fracture with linkage disequilibrium score regression and the MiXeR statistical tools.We used the conditional/conjunctional false discovery rate approach to identify the specific shared loci.We calculated the weighted genetic risk score for individuals in the UK Biobank and logistic regression was used to confirm the association observed in the prospective study.Results We found that MDD was associated with a 14%increase in fracture risk(hazard ratio(HR)1.14,95%CI 1.14 to 1.15,p<0.001)in the nested case-control analysis,while fracture was associated with a 72%increase in MDD risk(HR 1.72,95%CI 1.64 to 1.79,p<0.001)in the matched cohort analysis,suggesting a longitudinal and bidirectional relationship.Further,genetic summary data suggested a genetic overlap between MDD and fracture.Specifically,we identified four shared genomic loci,with the top signal(rs7554101)near SGIP1.The protein encoded by SGIP1 is involved in cannabinoid receptor type 1 signalling.We found that genetically predicted MDD was associated with a higher risk of fracture and vice versa.In addition,we found that the higher expression level of SGIP1 in the spinal cord and muscle was associated with an increased risk of fracture and MDD.Conclusions The genetic pleiotropy between MDD and fracture highlights the bidirectional association observed in the epidemiological analysis.The shared genetic components(such as SGIP1)between the diseases suggest that modulating the endocannabinoid system could be a potential therapeutic strategy for both MDD and bone loss.

Road Traffic Monitoring from Aerial Images Using Template Matching and Invariant Features

Road traffic monitoring is an imperative topic widely discussed among researchers.Systems used to monitor traffic frequently rely on cameras mounted on bridges or roadsides.However,aerial images provide the flexibility to use mobile platforms to detect the location and motion of the vehicle over a larger area.To this end,different models have shown the ability to recognize and track vehicles.However,these methods are not mature enough to produce accurate results in complex road scenes.Therefore,this paper presents an algorithm that combines state-of-the-art techniques for identifying and tracking vehicles in conjunction with image bursts.The extracted frames were converted to grayscale,followed by the application of a georeferencing algorithm to embed coordinate information into the images.The masking technique eliminated irrelevant data and reduced the computational cost of the overall monitoring system.Next,Sobel edge detection combined with Canny edge detection and Hough line transform has been applied for noise reduction.After preprocessing,the blob detection algorithm helped detect the vehicles.Vehicles of varying sizes have been detected by implementing a dynamic thresholding scheme.Detection was done on the first image of every burst.Then,to track vehicles,the model of each vehicle was made to find its matches in the succeeding images using the template matching algorithm.To further improve the tracking accuracy by incorporating motion information,Scale Invariant Feature Transform(SIFT)features have been used to find the best possible match among multiple matches.An accuracy rate of 87%for detection and 80%accuracy for tracking in the A1 Motorway Netherland dataset has been achieved.For the Vehicle Aerial Imaging from Drone(VAID)dataset,an accuracy rate of 86%for detection and 78%accuracy for tracking has been achieved.

A photogrammetric approach for quantifying the evolution of rock joint void geometry under varying contact states

Accurate measurement of the evolution of rock joint void geometry is essential for comprehending the distribution characteristics of asperities responsible for shear and seepage behaviors.However,existing techniques often require specialized equipment and skilled operators,posing practical challenges.In this study,a cost-effective photogrammetric approach is proposed.Particularly,local coordinate systems are established to facilitate the alignment and precise quantification of the relative position between two halves of a rock joint.Push/pull tests are conducted on rock joints with varying roughness levels to induce different contact states.A high-precision laser scanner serves as a benchmark for evaluating the photogrammetry method.Despite certain deviations exist,the measured evolution of void geometry is generally consistent with the qualitative findings of previous studies.The photogrammetric measurements yield comparable accuracy to laser scanning,with maximum errors of 13.2%for aperture and 14.4%for void volume.Most joint matching coefficient(JMC)measurement errors are below 20%.Larger measurement errors occur primarily in highly mismatched rock joints with JMC values below 0.2,but even in cases where measurement errors exceed 80%,the maximum JMC error is only 0.0434.Thus,the proposed photogrammetric approach holds promise for widespread application in void geometry measurements in rock joints.

TiN/Fe_(2)N/C composite with stable and broadband high-temperature microwave absorption

Facing the complex variable high-temperature environment,electromagnetic wave(EMW)absorbing materials maintaining high stability and satisfying absorbing properties is essential.This study focused on the synthesis and EMW absorbing performance evaluation of TiN/Fe_(2)N/C composite materials,which were prepared using electrostatic spinning followed by a high-temperature nitridation process.The TiN/Fe_(2)N/C fibers constructed a well-developed conductive network that generates considerable conduction loss.The heterogeneous interfaces between different components generated a significant level of interfacial polarization.Thanks to the synergistic effect of stable dielectric loss and optimized impedance matching,the TiN/Fe_(2)N/C composite materials demonstrated excellent and stable absorption performance across a wide temperature range(293-453 K).Moreover,TiN/Fe_(2)N/C-15 achieved a minimum reflection loss(RL)of−48.01 dB and an effective absorption bandwidth(EAB)of 3.64 GHz at 2.1 mm and 373 K.This work provides new insights into the development of high-efficiency and stabile EMW absorbing materials under complex variable high-temperature conditions.

Flexible ultrasound arrays with embossed polymer structures for medical imaging

In recent years,the application of flexible electronic technology[1−3]combined with ultrasonic imaging[4]has blossomed,resulting in a shift in the application place of ultrasonic imaging technology from medical institutions to households.The utilization of this application has expanded from the physician’s positioning and guidance to real-time patient monitoring[5],while the range of application scenarios continues to widen.The transformative changes have made possible through the development of flexible transducer arrays[6,7].Prof.Sheng Xu's team at the University of California San Diego has pioneered the development of various flexible transducer arrays based on island-bridge structures[8]and serpentine electrodes[9].These transducer arrays incorporate rigid 1−3 composite piezoelectric transducers on an island,featuring thin electrodes between the islands to provide mechanical flexibility,metallic electrodes of serpentine structure to provide stretchability of the device,and elastomer materials are used to encapsulate the overall structure,ensuring structural stability and impedance matching to the skin tissue[10].These flexible transducer arrays have found extensive applications in human blood pressure detection[11],cardiac imaging[12],blood flow doppler imaging[13],tissue modulus detection[14],and tissue deep hemoglobin detection[15].