Radial Moore graphs of radius three

The maximum number of vertices in a graph of specified degree and diameter cannot exceed the Moore bound. Graphs achieving this bound are called Moore graphs. Because Moore graphs are so rare, researchers have considered various relaxations of the Moore graph constraints. Since the diameter of a Moore graph is equal to its radius, one can consider graphs in which the condition on the diameter is relaxed, by one, while the condition on the radius is maintained. Such graphs are called radial Moore graphs. It has previously been shown that radial Moore graphs exist for all degrees when the radius is two. In this paper, we extend this result to radius three. We also construct examples that settle the existence question for a few new cases, and summarize the state of knowledge on the problem.     

A pH‐sensitive,strong double‐network hydrogel: Poly(ethylene glycol) methyl ether methacrylates–poly(acrylic acid)

We here describe new double network (DN) hydrogels with excellent mechanical strength and high sensitivity to pH changes. The first polymer network has a bottle brush structure and is formed from oligo‐monomers of poly(ethylene glycol) methyl ether methacrylate (PEGMA). Poly(acrylic acid) (PAA) is used as the second network. This double network features strong intermolecular interactions between the neutral poly(ethylene glycol) (PEG) side chains of PPEGMA and the non‐ionized carboxylic acid groups of the PAA second network. When immersed in solutions with a pH below ～4 the DN hydrogels have a low swelling ratio and are opaque as a result of solvent‐polymer phase separation driven by the formation of dense hydrogen‐bonded clusters. The compression strength (～8 MPa) is at least 14 times higher than the analogous single networks. When immersed in solutions with a pH >4, the hydrogels are transparent and exhibit a high swelling ratio with a compression strength of ～1 MPa. The PEG side chain length can be readily controlled without greatly altering the overall DN topology by choosing PEGMA monomers having different PEG side chain lengths. Longer PEG side branches give higher compression and tensile strengths at pH <4 when hydrogen bonded clusters form. The robust nature of these DN gels over a wide pH range may be useful for applications such as artificial muscles and controlled release devices. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Recursive constructions of small regular graphs of given degree and girth

We revisit and generalize a recursive construction due to Sachs involving two graphs which increases the girth of one graph and the degree of the other. We investigate the properties of the resulting graphs in the context of cages and construct families of small graphs using geometric graphs, Paley graphs, and techniques from the theory of Cayley maps.