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Given a graph Γ, a subset C of V (Γ) is called a perfect code in Γ if every vertex of Γ is at distance no more than one to exactly one vertex in C, and a subset C of V (Γ) is called a total perfect code in Γ if every vertex of Γ is adjacent to exactly one vertex in C. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.

Liquid–solid transition is a widely used strategy to shape polymeric materials and encode their microstructures. However, it is still challenging to fully exploit liquid behaviors of material precursors. In particular, the dynamic and static liquid behaviors naturally conflict with each other, which makes it difficult to integrate their advantages in the same materials. Here, by utilizing a shear-thinning phenomenon in the dynamic hybrid hydrogels, we achieve a hydrodynamic alignment of cellulose nanocrystals (CNC) and preserve it in the relaxed hydrogel networks due to the much faster relaxation of polymer networks (within 500 s) than CNC after the unloading of external force. During the following drying process, the surface tension of hydrogels further enhances the orientation index of CNC up to 0.872 in confined geometry, and these anisotropic microstructures demonstrate highly tunable birefringence (up to 0.004 14). Due to the presence of the boundaries of dynamic hydrogels, diverse xerogels including fibers, films, and even complex three-dimensional structures with variable anisotropic microstructures can be fabricated without any external molds.

Herein we report the fabrication of g-C3N4 nanofilms and their application as a solid fluorescence sensor. The as-prepared films are capable of convenient, sensitive, selective, rapid and recyclable sensing of Cu(2+) and Ag(+) in water and serum, indicating the sensor's potential application in disease diagnosis. Attractively, our sensor is able to differentiate Cu(2+) and Ag(+) by making use of their different adsorption and desorption kinetics during the interaction with g-C3N4 nanofilms.