In light of Lov & aacute;sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if the vertex-transitivity is relaxed, yet a high degree of symmetry-specifically edge-transitivity-is retained? To investigate this, we focus on the class of semisymmetric graphs, which are regular, edge-transitive, but not vertex-transitive. In this paper, it will be shown that every connected semisymmetric graph of order 2pq, where p and q are two distinct primes contains a Hamilton cycle and that every connected cubic semisymmetric graph of order less than 3000 contains a Hamilton cycle too. Based on these observations, the following question is posed: construct a connected semisymmetric graph which has no Hamilton cycle. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We investigate a class of singular quasilinear elliptic equations driven by double phase operators with variable coefficients. Under suitable assumptions on the parameter lambda, the coefficients, and the exponents, we employ the fibering method to analyze the variational structure of the associated energy functional on the Nehari manifold. More precisely, we derive an explicit sufficient threshold value Lambda and prove that the two components of the Nehari manifold are quantitatively separated whenever lambda < Lambda. This separation enables us to establish the existence of two distinct positive solutions, obtained as minimizers of the energy functional on the two components of the Nehari manifold. Furthermore, we derive explicit upper and lower bounds for both the L-p-norms of the gradients and the L alpha+1-norms of these solutions. Our estimates quantify the influence of the parameter lambda, the geometry of the domain, and the exponent alpha on the size of the corresponding solutions. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Two novel potassium manganese(II) oxalatophosphates, K2Mn(HPO4)(C2O4)center dot 2H2O and K2Mn(H2PO4)(C2O4)1.5, are successfully synthesized by a simple solvothermal reaction. Single crystal X-ray diffraction (SCXRD) analysis reveals that K2Mn(HPO4)(C2O4)center dot 2H2O and K2Mn(H2PO4)(C2O4)1.5 crystallize in the orthorhombic system (Pbca) and monoclinic system (P21/n). K2Mn(HPO4)(C2O4)center dot 2H2O presents a classical 2D layered structure, while K2Mn (H2PO4)(C2O4)1.5 contains 1D zig-zag chains. The crystal structures are verified by powder XRD, Fourier transform infrared (FTIR) and Raman spectra. Ultraviolet-visible-near-infrared (UV-vis-NIR) diffuse reflectance spectroscopy and density functional theory (DFT) calculations indicate that the band gaps are over 4.0 eV. Variable-temperature magnetization measurements confirm that both compounds contain high-spin Mn2+ ions and exhibit antiferromagnetic interactions.
In this work, nickel-doped ZnIn2S4 (xNi-ZIS) photocatalysts were prepared by a solvothermal method, and their light-driven hydrogen evolution and MO degradation performance were investigated. The optimized 2Ni-ZIS exhibits excellent photocatalytic performance, with a hydrogen evolution rate of up to 20.74 mmol g-1 h-1, more than 8 times that of ZIS. The degradation rate of MO reached 81.8% within 90 min. Further magnetization of the 2Ni-ZIS sample (2Ni-ZIS-Mag) significantly enhances its hydrogen evolution performance to 30.19 mmol g-1 h-1, with an apparent quantum efficiency of 6.74% at 420 nm. The MO degradation rate increases to 97.2%. The experimental results indicate that Ni doping causes a red shift of the absorption edge of ZnIn2S4, markedly improving its visible-light absorption capability. Electron paramagnetic resonance and field cooling demonstrate that Ni doping induces a magnetic moment of 2.83 mu B in ZnIn2S4. This is attributed to the optimization of the electron spin configuration in 2Ni-ZIS-Mag. Its carrier lifetime is extended from 25.68 ns to 31.36 ns, and the effective magnetic moment is also increased to 3.47 mu B. The induced localized spin-polarized state effectively promotes the separation and migration of photogenerated carriers, thereby prolonging the carrier lifetime. Theoretical analysis further reveals that the excellent photocatalytic performance of Ni-ZIS also originates from the elevated Fermi level and optimized hydrogen evolution free energy. Based on mass spectrometry and the excited-state Fukui function, the possible degradation pathways of the MO molecule were proposed
Developing electrocatalysts that combine high activity, industrial-level current densities, and excellent durability remains a major challenge for the practical application of water electrolysis. In this work, a three-dimensional porous NiCoFe alloy electrode was fabricated via a bubble-template electrodeposition strategy. The porous structure effectively promotes gas release and efficient electrolyte transport. The incorporation of Fe enables efficient synergy between the adsorbate evolution mechanism (AEM) and the lattice oxygen mechanism (LOM), thereby accelerating the oxygen evolution reaction (OER) kinetics. Benefiting from the synergistic regulation of structural and electronic properties, the NiCoFe electrode achieves industrial-scale current densities of 1000 and 2000 mA cm-2 in 1 M KOH electrolyte with overpotentials of only 293 mV and 334 mV, respectively. When assembled as the anode in an anion exchange membrane water electrolyzer (AEMWE), the device delivers a current density of 1 A cm-2 at a cell voltage of only 1.76 V at 60 degrees C and operates stably for 1000 h. Density functional theory (DFT) calculations reveal that Fe incorporation elevates the O 2p band center and weakens the M-O bond strength, thereby facilitating lattice oxygen activation and optimizing the adsorption behavior of key reaction intermediates. This work provides a simple and effective alloying strategy for tuning catalytic properties and offers new insights into the design of efficient and durable OER electrocatalysts.
