In light of Lovász'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.
This paper is concerned with the existence and stability of a type of blowing-up positive steady states for a shadow system of the Shigesada–Kawasaki–Teramoto (SKT) two-species competition model and for the perturbed SKT system with a sufficiently large cross-diffusion parameter and bounded random diffusion parameters. The asymptotic behavior of coexistence steady states of the SKT model as one of the cross-diffusion parameters tends to infinity was studied by Lou and Ni [1], who showed that almost all coexistence states can be characterized by one of three shadow systems. In [2], the first author of the present paper studied one of these shadow systems with one cross-diffusion term in one space dimension and proved that one component along a bifurcation branch blows up as the bifurcation parameter approaches the least positive eigenvalue of −Δ under the homogeneous Neumann boundary condition. Using a different approach from that in [2], based on suitable transformations and the Lyapunov-Schmidt reduction, we establish the existence and detailed asymptotic structure of several branches of positive steady states near the blow-up points for shadow systems with one or two cross-diffusion terms in one- and multi-dimensional domains. We further prove that all the large steady states obtained near the blow-up points are spectrally unstable. Finally, by perturbation arguments, we establish the existence and instability of corresponding branches of positive steady states for the original SKT system when one of the cross-diffusion parameters is sufficiently large.
In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces Hs(Rd) for all s > 0 and d=2,3. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields B0∈Rd, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.
Perylene diimide (PDI) radical anions are attractive for near-infrared (NIR) photothermal applications, but strong aggregation in aqueous media severely limits their biomedical utility. Here, we report an extended tetrapodal PDI dye, PDI-4PPy-14+, featuring four elongated 4-phenyl-pyridinium pendants that provide substantial steric hindrance and significantly suppress chromophore aggregation. This structural design stabilizes PDI radical anions under reductive conditions and enables efficient NIR photothermal conversion. Compared with conventional PDIs, PDI-4PPy-14+ exhibits markedly enhanced bio-reduction-driven photothermal activity, allowing selective and effective antibacterial treatment. These findings demonstrate the critical role of deliberate structural tailoring in modulating aggregation and functional performance of organic dyes, offering a general strategy for designing non-aggregating, highly isolated chromophores for targeted biomedical photothermal applications.
In this paper, we study the dot product graphs in F-q(d). We prove that if the size of the product of two adjacent sets is large enough, then the set of dot product graphs has positive density. Our method is based on finite field Fourier analytic techniques. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
