The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given. For a connected $k$-uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if $k$ is even and the hypergraph is odd-bipartite.Comment: 26 pages, 3 figure

Epoxy/Polycaprolactone Systems with Triple-Shape Memory Effect: Electrospun Nanoweb with and without Graphene Versus Co-Continuous Morphology

Triple-shape memory epoxy (EP)/polycaprolactone (PCL) systems (PCL content: 23 wt %) with different structures (PCL nanoweb embedded in EP matrix and EP/PCL with co-continuous phase structure) were produced. To set the two temporary shapes, the glass transition temperature (Tg) of the EP and the melting temperature (Tm) of PCL served during the shape memory cycle. An attempt was made to reinforce the PCL nanoweb by graphene nanoplatelets prior to infiltrating the nanoweb with EP through vacuum assisted resin transfer molding. Morphology was analyzed by scanning electron microscopy and Raman spectrometry. Triple-shape memory characteristics were determined by dynamic mechanical analysis in tension mode. Graphene was supposed to act also as spacer between the nanofibers, improving the quality of impregnation with EP. The EP phase related shape memory properties were similar for all systems, while those belonging to PCL phase depended on the structure. Shape fixity of PCL was better without than with graphene reinforcement. The best shape memory performance was shown by the EP/PCL with co-continuous structure. Based on Raman spectrometry results, the characteristic dimension of the related co-continuous network was below 900 nm