Resistor Network Research Articles

Resistor Network for Forward Calculation in Electrical Resistance Tomography of Carbon Fiber-Reinforced Composites

Resistor Network for Forward Calculation in Electrical Resistance Tomography of Carbon Fiber-Reinforced Composites

Antifreezing functionalized-alginate-based electrolytes for zinc-ion batteries

Flexible zinc-ion batteries (ZIBs) assembled with hydrogel electrolyte are considered as promising flexible energy storage devices because of their inherent safety and versatility. However, the ionic conductivity and mechanical properties of most hydrogel electrolytes are not satisfactory, Furthermore, they will freeze at subzero temperature due to existing water. In this work, a freezing resistant polycarboxylic double network gel electrolyte (SIP-CS) with high ionic conductivity (14.36 mS cm⁻1 at −20 °C) and excellent mechanical property (fracture stress of 241.5 kPa and fracture strain of 1011 %) is prepared. Iminodiacetic acid (IDA) is applied to modify the alginate mainchains with many -COOH groups, which could provide channels for ion migration and endow hydrogel with high ionic conductivity. Additionally, sorbitol, containing lots of hydroxyl groups, is applied as a cryoprotectant to enhance the subzero performance of the electrolyte, because sorbitol could break the hydrogen bonds between water molecules, inhibit the formation of ice crystals, and reduce the freezing point of the gel electrolyte. The freezing point of SIP-CS is −37.0 °C, enabling the electrolyte to perform well at low temperatures. The flexible quasi solid Zn-MnO2 battery is assembled to evaluate the low-temperature electrochemical performance of SIP-CS. The assembled Zn-MnO2 battery shows good cycling and stable electrochemical performance, which proves the excellent antifreezing property of SIP-CS. This work provides a new strategy for preparing an electrolyte with good withstand low-temperature capability.

Leading two-point resistances from transfer matrices in cylindrical, spider web, axial and grid resistor networks

Resistor networks are increasingly being considered in heuristic research as models for natural or artificial matter. The equivalent resistance between two nodes, the Two-Point Resistance (TPR), can be calculated using a variety of methods. The transfer matrix (TM) method was originally considered as a numerical tool for estimating percolation thresholds in random networks with a repeating pattern. The TM method is revisited here as an efficient tool to obtain, in a fast and elegant way, iteration relations and exact explicit expressions for leading TPRs that include a node in the last repeated pattern. Several rotationally invariant networks are studied, such as simple cylindrical networks, spider web networks and cylindrical networks with a central resistive axis, in which case the TM matrices are circulant matrices. Examples of explicit expressions are given for orders of rotation ≤4 or 5, depending on the case. The method can be applied in a similar way to networks with less symmetry, such as grids. The general expressions of TPRs obtained using the TM method can provide quantitative guidelines for resistor networks developed in materials science, environmental issues or industrial applications.

Fully coupled thermal-structural-vibration characteristics of 3-axis machine tools with complex hybrid linear axis-spindle systems considering structural nonlinearities

For machine tools composed of functional components bonded by joints with structural nonlinearities, the load- and temperature-dependent nonlinear restoring force,stiffness, and frictional heat generationfrom the jointdetermine its complex dynamic fully coupled thermal-structural-vibration characteristics. In response, in this work, considering the vibration feedback dominated by the structural vibration and employing a partition-based closed-loop iterative algorithm, a fully coupled thermal-structural-vibration model that characterizes the nonlinear coupling between temperature and mechanical fields is developed with the joint as the link. The multi-body dynamics model of the whole machine tool, considering the flexibility of the screw shaft, spindle, and joint nonlinearity, characterizes the milling dynamics influenced by structural vibration and regeneration effects. The global thermal resistance network model with dynamic heat flux, viscosity-temperature effect, and time-varying thermal resistanceis establishedto characterize the thermal characteristics. The partition-based closed-loop iterative algorithm is employed to realize the multi-field coupling and consider the moving heat source and the load distribution change caused by the reciprocating moving joint. Considering the changes in contact angle and contact deformation caused by preload, dynamic load, and thermal effect, load- and temperature-dependent nonlinear joint restoring force and stiffnessare developed. Based on the viscosity-temperature effects and the change of contact friction caused by thermal expansion and dynamic load, the load- and temperature-dependent nonlinear joint frictional heat generation is calculated. The milling load from tool-workpiece interaction is calculated, and the influence of multi-field coupling on regenerative chatter is explored. The proposed model was experimentally verified, andthe effectof parameterswas numerically investigated. The results show that the dynamic milling load excites the whole machine tool vibration, and for every 10 μm increase in cutting depth, the radial amplitude of the tool nose increased by at least 1 μm. The nonlinear coupling between temperature and mechanical field significantly affects the system response, and the joint contact stiffness dominates the multi-field coupling behavior.

Unraveling the mechanisms of glioblastoma's resistance: investigating the influence of tumor suppressor p53 and non-coding RNAs.

Glioblastoma (GB) is one of the most fatal CNS malignancies, and its high resistance to therapy and poor outcomes have made it one of the primary challenges in oncology. Resistance to standard therapy, i.e., radio-chemotherapy with temozolomide, is one of the principal causes of the poor prognostic outcomes of GB. Finding the molecular basis of GB resistance to therapy is key to creating effective solution approaches. The general problem of GB resistance is supervised by cancer suppressive protein, p53, and has become a very special interest in molecular research in recent decades. The principal aim of this manuscript is to perform a comprehensive survey on the complex network of interactions developed by p53 with non-coding RNAs (ncRNA) in the context of GB resistance. The present article details the functional aspects of p53 as a cellular stress response protein, including its roles in apoptosis, cell cycle regulation, and DNA repair in glioblastoma (GB), along with the disruption of p53 and its involvement in chemoresistance (CR). It also highlights several classes of ncRNAs, namely microRNAs, long ncRNAs, and circular RNAs, that manipulate p53 signaling in GB-CR. The article likewise explains how disruption in the expression of these ncRNAs can promote GB-CR and how it interacts with essential cellular functions, such as proliferation, apoptosis, and DNA repair. The manuscript also describes the potential of targeting p53 and ncRNAs with their diagnostic and prognostic potential as novel promising therapeutics for GB. Nevertheless, ncRNA-based biomarkers still present challenges for their suitability in GB resistance. However, modern research continues to discover novel prediction targets, potentially enhancing patient outcomes and therapeutic options. Therefore, the neutralization of this intricate regulatory network of GB resistance might have a primary clinical effect in fighting GB resistance therapy and thus might lead to a substantial increase in patient survival and quality of life.

Parametrical synthesis of various radio devices with the set quantity of identical cascades of type «the nonlinear part – the mixed two-port network»

Background. Presence of possibility of analytical definition of a part of parametres of various radio devices, optimum by criterion of maintenance of preset values of modules and phases of transfer functions on necessary quantity of frequencies, considerably reduces time of numerical optimisation of other part of parametres by criterion of formation demanded frequency response and phase response in a strip of frequencies. Till now such problems dared concerning radio devices only with one cascade of type «a nonlinear part - the coordination the device» or «the coordination the device – a nonlinear part». In quality «the coordination devices were used the jet, resistive, complex or mixed two-port networks. The problem of multicascade radio devices with jet two-port networks is solved also. Change of basis for the coordination two-port networks and a place of inclusion of a nonlinear part leads to change of area of a physical realizability. Aim. Working out of algorithms of parametrical synthesis of radio devices with any quantity of identical and unequal cascades of type «a nonlinear part ‑the coordination the mixed two-port network» by criterion of maintenance of the set frequency characteristics. Nonlinear parts are presented in the form of a nonlinear element and parallel either consecutive on a current or pressure of a feedback. Methods. The theory of two-port networks, matrix algebra, a decomposition method, a method of synthesis of actuation microwave devices, numerical methods of optimisation. Results. In interests of achievement of the specified purpose systems of the algebraic equations are generated and solved. Models of optimum two-port networks in the form of mathematical expressions for definition of interrelations between elements of their classical matrix of transfer and for search of dependences of mixed of two-poles from frequency are received. It is shown, that at certain parities between quantity of cascades and values of resistance of a source of a signal and loading of the one-cascade radio device frequency characteristics of one-cascade and multicascade radio devices appear identical or similar. Such schemes are named by equivalent. Conclusion. The comparative analysis of theoretical results (frequency response and phase response radio devices, value of parametres), received by mathematical modelling in system MathCad, and the experimental results received by схемотехнического of modelling in systems OrCad and MicroCap, shows their satisfactory coincidence.

Gene co-expression analysis reveals conserved and distinct gene networks between resistant and susceptible Lens ervoides challenged by hemibiotrophic and necrotrophic pathogens

As field crops are likely to be challenged by multiple pathogens during their development, the investigation of broad-spectrum resistance in the host is of great interest for crop genetic enhancement. In this study, we attempted to address this question by adopting a weighed gene co-expression approach to study the temporal transcriptome dynamics of resistant and susceptible recombinant inbred lines (RILs) derived from an intraspecific Len ervoides cross during the infection process with either the necrotrophic pathogens Ascochyta lentis or Stemphylium botryosum, or the hemibiotrophic pathogen Colletotrichum lentis. By comparing networks of resistant and susceptible RILs, seven network module pairs were found to possess high correlation coefficients (R > 0.70) and large number of overlapping genes (n > 100). The conserved co-regulation of genes in these network module pairs were involved in plant cell wall synthesis, cell division, cytoskeleton organization, and protein ubiquitin related processes and appeared to be common disease responses against these pathogens. On the other hand, we also identified eight modules with low correlation between resistance and susceptibility networks. Among those, a stronger gene co-expression in R-genes and small RNA processes in the resistant hosts may be enhancing L. ervoides resistance against A. lentis, C. lentis, and S. botryosum, whereas the higher level of synergistic regulation in the synthesis of arginine and glutamine and phospholipid and glycerophospholipids in the susceptible hosts may contribute to increased susceptibility in L. ervoides.

Modeling of adsorption-controlled binary gas transport in ultratight porous media

Organic-rich ultratight porous media such as shales have complicated pore types and sizes, dictating their unique fluid transport and storage mechanisms. This paper introduces a diffusion-based binary gas (CH4 and CO2) transport model in such porous media, incorporating key transport and storage mechanisms. Binary gas mixture within the pore volume is divided into two domains: the free-phase and the sorbed-phase, both of which contribute to gas mass transport. Thermodynamic properties of the free phase are determined using the Peng-Robinson equation of state, while a modified extended Langmuir isotherm describes the sorbed phase. The bulk diffusion, Knudsen diffusion, and viscous diffusion are reconciled to describe binary gas transport in the free phase, while the surface diffusion is considered in the sorbed phase. The mass flux is finally related to the effective diffusion coefficient and the free-phase concentration gradient through coupling all transport mechanisms via the equivalent resister network model. The governing equations are solved numerically using a finite difference approach to simulate co-diffusion and counter-diffusion of a binary gas mixture in two nanoporous media, shale and coal.The results reveal that surface diffusion is more pronounced in coal than in shale due to coal's higher adsorption capacity. In co-diffusion cases, both free-phase and sorbed-phase concentrations decrease over time. However, the decline of sorbed-phase concentrations is slower than that of the free-phase. Moreover, CO2 tends to remain adsorbed within the coal matrix due to its stronger adsorption affinity, unlike CH4, which is more likely to flow out. In counter-diffusion cases, injected CO2 first accumulates near the fracture region and then diffuses further into the matrix. Despite shale's pore volume being twice that of coal, similar amounts of CO2 are injected, which can be attributed to coal's higher CO2 adsorption capacity.

Transmitted drug resistance and molecular transmission network among treatment-naive HIV-1 patients in Wenzhou, China, 2020–2023

BackgroundTransmitted drug resistance (TDR) increases the risk of antiretroviral therapy (ART) failure in HIV-1 patients. This study investigated the molecular epidemiology of TDR and its transmission networks among newly diagnosed HIV-1 patients in Wenzhou, China.MethodsWe enrolled 1878 ART-naive HIV-1 patients from January 2020 to October 2023. TDR was evaluated using the Stanford University HIV Drug Resistance Database. We performed phylogenetic analysis, genotyping, transmission clustering, and population-based TDR-related factor analysis.ResultsAmong 1782 patients with successful genotyping, TDR prevalence was 5.7%. Multivariable analysis identified CRF08_BC subtype (adjusted odds ratio [aOR] 18.59, 95% CI 3.79-336.18, p = 0.004), CD4 > 500 cells/mm³ (aOR 2.19, 95% CI 1.16–4.03, p = 0.013), and year 2023 (aOR 1.83, 95% CI 1.11–4.89, p = 0.039) as factors associated with higher TDR risk. The most prevalent NNRTI mutations were K103N, E138A, and V179E. Seven TDR transmission clusters were identified, notably one with V179D that expanded during 2020–2023.ConclusionsWhile TDR prevalence in Wenzhou remained lower than in other Chinese regions, an upward trend was observed. Most resistant individuals were in transmission clusters, predominantly middle-aged and elderly. NNRTI resistance was severe and concentrated in efavirenz, nevirapine, and rilpivirine. Enhanced HIV surveillance and wider free antiretroviral options are crucial to control drug-resistant HIV spread in Wenzhou.

Quantitative Trait Locus Mapping Combined with RNA Sequencing Identified Candidate Genes for Resistance to Powdery Mildew in Bitter Gourd (Momordica charantia L.)

Improving the powdery mildew resistance of bitter gourd is highly important for achieving high yield and high quality. To better understand the genetic basis of powdery mildew resistance in bitter gourd, this study analyzed 300 lines of recombinant inbred lines (RILs) formed by hybridizing the powdery mildew-resistant material MC18 and the powdery mildew-susceptible material MC402. A high-density genetic map of 1222.04 cM was constructed via incorporating 1,996,505 SNPs generated by resequencing data from 180 lines, and quantitative trait locus (QTL) positioning was performed using phenotypic data at different inoculation stages. A total of seven QTLs related to powdery mildew resistance were identified on four chromosomes, among which qPm-3-1 was detected multiple times and at multiple stages after inoculation. By selecting 18 KASP markers that were evenly distributed throughout the region, 250 lines and parents were genotyped, and the interval was narrowed to 207.22 kb, which explained 13.91% of the phenotypic variation. Through RNA-seq analysis of the parents, 11,868 differentially expressed genes (DEGs) were screened. By combining genetic analysis, gene coexpression, and sequence comparison analysis of extreme materials, two candidate genes controlling powdery mildew resistance in bitter gourd were identified (evm.TU.chr3.2934 (C3H) and evm.TU.chr3.2946 (F-box-LRR)). These results represent a step forward in understanding the genetic regulatory network of powdery mildew resistance in bitter gourd and lay a molecular foundation for the genetic improvement in powdery mildew resistance.

Advancements in Flexible and Stretchable Electronics for Resistive Hydrogen Sensing: A Comprehensive Review.

Flexible and stretchable electronics have emerged as a groundbreaking technology with wide-ranging applications, including wearable devices, medical implants, and environmental monitoring systems. Among their numerous applications, hydrogen sensing represents a critical area of research, particularly due to hydrogen's role as a clean energy carrier and its explosive nature at high concentrations. This review paper provides a comprehensive overview of the recent advancements in flexible and stretchable electronics tailored for resistive hydrogen sensing applications. It begins by introducing the fundamental principles underlying the operation of flexible and stretchable resistive sensors, highlighting the innovative materials and fabrication techniques that enable their exceptional mechanical resilience and adaptability. Following this, the paper delves into the specific strategies employed in the integration of these resistive sensors into hydrogen detection systems, discussing the merits and limitations of various sensor designs, from nanoscale transducers to fully integrated wearable devices. Special attention is paid to the sensitivity, selectivity, and operational stability of these resistive sensors, as well as their performance under real-world conditions. Furthermore, the review explores the challenges and opportunities in this rapidly evolving field, including the scalability of manufacturing processes, the integration of resistive sensor networks, and the development of standards for safety and performance. Finally, the review concludes with a forward-looking perspective on the potential impacts of flexible and stretchable resistive electronics in hydrogen energy systems and safety applications, underscoring the need for interdisciplinary collaboration to realize the full potential of this innovative technology.

Heat transfer performance and startup characteristics of separated gravity heat pipe

Separated gravity heat pipes, with distinct arrangements of evaporating and condensing sections, offer low thermal resistance, high heat transfer efficiency, and a simple structure, making them suitable for passive cooling of spent fuel pools through gravity-driven processes. In this study, a steady-state, one-dimensional, two-phase flow model is developed to analyze phase change heat transfer in a separated gravity heat pipe. The research investigates the impact of liquid filling rate and vacuum level on heat pipe operation. The findings reveal a notable increase in pressure drop inside the tube with higher liquid filling rates, which diminishes the adaptability of the heat pipe as pressure rises, resulting in a narrower range of effective liquid filling rates. Specifically, the liquid filling rate of the heat pipe ranges from 38 % to 72 % at P = 13.75 kPa and from 44 % to 68 % at P = 15.75 kPa. Additionally, a startup prediction model is formulated based on a thermal resistance network model using the ideal gas assumption. Two startup states—namely, the initial startup state and the subsequent startup state—are proposed for large-scale separated gravity heat pipe arrays. The heat pipe achieves the large-temperature-difference initial startup state at a 50 % liquid filling rate within 7 min before transitioning to the subsequent startup state. The vapor mass, heat transfer coefficient, and working fluid temperature initially increase and then decrease over time, with varying peak times. As the liquid filling rate increases, the maximum vapor mass also increases. The heat transfer coefficient peaks at a 60 % liquid filling rate before subsequently declining. The optimal liquid filling rate of 60 % is identified based on startup time, heat transfer coefficient, and overall heat transfer efficiency.

Analytical Electro-Thermal Model and 3-D Thermal Resistance Network for SMD-Based Printed Circuit Board Power Converters

Analytical Electro-Thermal Model and 3-D Thermal Resistance Network for SMD-Based Printed Circuit Board Power Converters

Micro-Channel Cooling of Hot Spots Through Non-Uniform Aspect Ratio Designs

Abstract Electronic devices experience spatial variation in power dissipation, which results in high-temperature hot spots. These locations require aggressive thermal management, which can be complex and costly. Simple solutions such as single-phase micro-channels can provide adequate heat transfer, but they are not designed to control heat transfer locally. However, micro-channels can be tailored to control local flow rates and heat transfer, potentially mitigating hot spot temperatures. Using a conductive and convective resistance network for a micro-channel, an analytical model is generated for heat transfer within an individual passage. For a given channel width, this model relates the channel depth to its resistance through a power law. Over a wide range of heat fluxes, the optimal design balances local temperatures to within 3 K. The analytical model is validated using computational simulations of the optimized heat sink. For a randomly-generated, non-uniform power distribution, device temperatures are balanced with a sample standard deviation below 2.5%, which is significantly better than a baseline design. When heat spreading is incorporated, the temperature increase is smaller but remains uniform, indicating that the hot spots can be mitigated.

The Seven-State RF MEMS Miniaturized Broadband Reconfigurable Step Attenuator.

This paper presents a three-channel reconfigurable step attenuator based on radio frequency (RF) microelectromechanical system (MEMS) switches, in response to the current issues of high insertion loss and low attenuation accuracy of attenuators. The coplanar waveguide (CPW), cross-shaped power dividers, RF MEMS switches, and π-type attenuation resistor networks are designed as a basic unit of the attenuator. The attenuator implemented attenuation of 0~30 dB at 5 dB intervals in the frequency range of 1~25 GHz through two basic units. The results show that the insertion loss is less than 1.41 dB, the attenuation accuracy is better than 2.48 dB, and the geometric size is 2.4 mm × 4.0 mm × 0.7 mm. The attenuator can be applied to numerous fields such as radar, satellites, aerospace, electronic communication, and so on.

An in vitro set-up to study Pdr5-mediated substrate translocation.

Pdr5 is the most abundant ABC transporter in Saccharomyces cerevisiae and plays a major role in the pleiotropic drug resistance (PDR) network, which actively prevents cell entry of a large number of structurally unrelated compounds. Due to a high level of asymmetry in one of its nucleotide binding sites (NBS), Pdr5 serves as a perfect model system for asymmetric ABC transporter such as its medical relevant homologue Cdr1 from Candida albicans. In the past 30 years, this ABC transporter was intensively studied in vivo and in plasma membrane vesicles. Nevertheless, these studies were limited since it was not possible to isolate and reconstitute Pdr5 in a synthetic membrane system while maintaining its activity. Here, the functional reconstitution of Pdr5 in a native-like environment in an almost unidirectional inside-out orientation is described. We demonstrate that reconstituted Pdr5 is capable of translocating short-chain fluorescent NBD lipids from the outer to the inner leaflet of the proteoliposomes. Moreover, this transporter revealed its ability to utilize other nucleotides to accomplish transport of substrates in a reconstituted system. Besides, we were also able to estimate the NTPase activity of reconstituted Pdr5 and determine the kinetic parameters for ATP, GTP, CTP, and UTP.

Understanding disorder in monolayer graphene devices with gate-defined superlattices

Engineering superlattices (SLs)-which are spatially periodic potential landscapes for electrons-is an emerging approach for the realization of exotic properties, including superconductivity and correlated insulators, in two-dimensional materials. While moiré SL engineering has been a popular approach, nanopatterning is an attractive alternative offering control over the pattern and wavelength of the SL. However, the disorder arising in the system due to imperfect nanopatterning is seldom studied. Here, by creating a square lattice of nanoholes in the SiO2dielectric layer using nanolithography, we study the SL potential and the disorder formed in hBN-graphene-hBN heterostructures. Specifically, we observe that while electrical transport shows distinct SL satellite peaks, the disorder of the device is significantly higher than graphene devices without any SL. We use finite-element simulations combined with a resistor network model to calculate the effects of this disorder on the transport properties of graphene. We consider three types of disorder: nanohole size variations, adjacent nanohole mergers, and nanohole vacancies. Comparing our experimental results with the model, we find that the disorder primarily originates from nanohole size variations rather than nanohole mergers in square SLs. We further confirm the validity of our model by comparing the results with quantum transport simulations. Our findings highlight the applicability of our simple framework to predict and engineer disorder in patterned SLs, specifically correlating variations in the resultant SL patterns to the observed disorder. Our combined experimental and theoretical results could serve as a valuable guide for optimizing nanofabrication processes to engineer disorder in nanopatterned SLs.

Design of ant colony planning algorithms for multi-stage grids of distribution networks considering network resilience

Network Destruction Resilience is the ability of a system to maintain good operation in the event of an external attack or internal failure. The resilience of distribution network is crucial to guarantee the reliability of power supply. In this paper, we design a planning algorithm considering network destructibility for the multi-stage ant colony planning problem of distribution networks. The method establishes the multi-stage network planning objective function of distribution network from the perspectives of total investment cost and annual operation cost of multi-node network frame, destruction resistance, active power and reactive network loss of distribution network, etc. Then based on the constraints of the model, the improved ant colony algorithm is used to solve the multi-stage network planning objective function of distribution network, and the results of the anti-termite colony planning of multi-stage network frame of distribution network are obtained. In order to verify the effectiveness of the algorithm, simulation experiments are carried out on real distribution network data. The results show that the proposed ant colony planning algorithm for multi-stage grid frames can effectively improve the destruction resistance of distribution networks, and reduce the total investment cost and annual operation cost of multi-stage grid frames, and reduce the network loss rate data of multi-stage grid frames of distribution networks after application. It provides an effective method for planning the distribution network.

A novel formula for representing the equivalent resistance of the m×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ imes n$$\\end{document} cylindrical resistor network

The problem of solving the equivalent resistance between two points for resistor networks has important significance in physics. This paper mainly changes and rewrites the formula for calculating the resistance between two points of an unconventional m×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ imes n$$\\end{document} cylindrical resistor network with a zero resistor axis and any two left and right boundaries. To enhance the efficiency of calculating the equivalent resistance between two points, Chebyshev polynomials and hyperbolic cosine functions are employed to represent the new formula. And in the inference process, the famous discrete cosine transform of the third kind (DCT-III) is used to process the matrix. We give the equivalent resistance formula for several special cases, and display them by a three-dimensional graph. Subsequently, the calculation efficiency of the original formula and the rewritten formula are compared. At the end of the paper, a heuristic algorithm suitable for robot path planning on cylindrical environment is proposed.

Optimal resistor networks

AbstractGiven a graph on vertices with edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions — graphs like a star, and graphs which are close to regular — with the transition between them occurring when the average degree is 3. However, in this paper, we show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs — namely ‘which rooted graph minimises the average resistance to the root?’. The rooted case is much simpler to analyse that the unrooted, and the one of the main results of this paper is that the two cases are asymptotically equivalent.