Rapid synthesis of the low thermal expansion phase of Al<Subscript>2</Subscript>Mo<Subscript>3</Subscript>O<Subscript>12</Subscript> via a sol–gel method using polyvinyl alcohol

Aluminum molybdate was successfully synthesized using a simplified PVA assisted sol–gel method resulting in highly crystalline, monophasic (monoclinic P2₁/a) samples. These materials could readily be obtained at temperatures of 600 and 700 °C after calcining for as little as 15–20 min. Scanning electron microscopy and X-ray powder diffraction indicated that even the sample calcined at 600 °C for 20 min was free of impurities and composed of submicron sized particles (~300 nm). Transmission electron microscopy was used to confirm the monophasic character and submicron dimensions of the as-prepared powders. In addition to producing high quality samples, it was also observed that the metal to PVA ratio used during this simplified synthesis, could be used as a control parameter for tailoring the particle sizes of the final product.     

Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of $G$ act as automorphisms of the deep commuting graph.