ABSTRACT:The resistance distance r ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf (G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L n consists of the Laplacian spectrum of path P 2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed-form formula for Kirchhoff index of L n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L n is approximately to one half of its Wiener index. Finally, we show thatholds for all graphs G in a class of graphs including L n .

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ABSTRACT:The resistance distance r ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf (G) is the sum of resistance distances between all pairs of vertices. In this work, closed-form formulae for Kirchhoff index and resistance distances of circulant graphs are derived in terms of Laplacian spectrum and eigenvectors. Special formulae are also given for four classes of circulant graphs complete graphs, complete graphs minus a perfect matching, cycles, Möbius ladders M p . In particular, the asymptotic behavior of Kf

Na-ion batteries (NIBs) have attracted increasing attention given the fact that sodium is relatively more plentiful and affordable than lithium for sustainable and large-scale energy storage systems. However, the shortage of electrode materials with outstanding comprehensive properties has limited the practical implementations of NIBs. Among all the discovered anode materials, transition-metal sulfide has been proven as one of the most competitive and promising ones due to its excellent redox reversibility and relatively high theoretical capacity. In this study, double-morphology N-doped CoS/multichannel carbon nanofibers composites (CoS/MCNFs) are precisely designed, which overcome common issues such as the poor cycling life and inferior rate performance of CoS electrodes. The conductive 3D interconnected multichannel nanostructure of CoS/MCNFs provides efficient buffer zones for the release of mechanical stresses from Na ions intercalation/deintercalation. The synergy of the diverse structural features enables a robust frame and a rapid electrochemical reaction in CoS/MCNFs anode, resulting in an impressive long-term cycling life of 900 cycles with a capacity of 620 mAh g at 1 A g (86.4% theoretical capacity) and a surprisingly high-power output. The proposed design in this study provides a rational and novel thought for fabricating electrode materials.

Separators are vital parts of all lithium-ion batteries (LIBs), and they play a critical role in their thermal safety and electrochemical performance. Considering the low thermal stability and inferior electrolyte wettability of commercial polyolefin membranes, in this study, a novel thermotolerant polyimide aerogel (PIA) separator is designed. This is the first report on the use of chemically cross-linked PIA separators for LIBs. The outstanding porosity (78.35%) and electrolyte absorption (321.66%) of PIA separators contribute to the low internal resistance and outstanding electrochemical performance of LIBs, which can retain a high specific capacity of 118 mAh g −1 after 1000 cycles at a current density of 1 C. In addition, The LiFePO 4 |Li metal batteries with PIA separators are extremely stable at 90 °C (>300 cycles), and maintain stability even at 120 °C. More importantly, for pouch batteries, PIA separators result in higher thermal runaway temperatures compared with Celgard separators. In this paper, the pyrolysis mechanism and thermodynamic process of PIA separators are clarified by DFT calculations and in situ synchrotron vacuum UV photoionization mass spectrometry experiments.