Undirected power graphs of semigroups

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a ^m =b or b ^m =a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p ^m . Particular attention is given to the multiplicative semigroup ℤ _n and its subgroup U _n , where G(U _n ) is a major component of G(ℤ _n ). It is proved that G(U _n ) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U _n ) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U _n ) has no Hamiltonian cycle.     

Modulation of Biophysical Cues in Nature Inspired Patterning of Porous Silk Fibroin Scaffold for Replenishable Controlled Drug Delivery

While a sticking plasteris enough for healing of most of the minor cuts they may get routinely, critical situations like surgical, gunshot, accidental or diabetic wounds;lacarations and other cutaneous deep cuts may require implants and simultaneous medications for healing. From the biophysical standpoint, an internal force-based physical surface stimulusis crucial for cellular sensing during wound repair. In this paper, the authors report the fabrication of a porous, biomimmetically patterned silk fibroin scaffold loaded with ampicillin, which exhibits controlled release of the drug along with possible replenishment of the same. In vitro swelling study reveals that the scaffolds with hierarchical surface patterns exhibit lower swelling and degradation than other types of scaffolds. The scaffolds, that show remarkable broad–spectrum antibacterial efficacy, exhibit Korsemeyer–Peppas model for the ampicillin release patterns due to the structural hydrophobicity imparted by the patterns. Four distinct cell-matrix adhesion regimes are investigated for the fibroblasts to eventually form cell sheets all over the hierarchical surface structures. 4',6–diamidino–2–phenylindole (DAPI) and Fluorescein Diacetate (FDA) fluorescent staining clearly demonstrate the superiority of patterned surface over its other variants. A comparative immunofluorescence study among collagen I, vinculin, and vimentin expressions substantiated the patterned surface to be superior to others.