Certain Classes of Cohen-Macaulay Multipartite Graphs

The Cohen-Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen-Macaulay property of a graph arising from a family of reflexive and antisymmetric relations on a set. We use this result to find classes of multipartite graphs which are Cohen-Macaulay

Modules with Pure Resolutions

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.Comment: 9 page

X-ray Diffraction Analysis of Cu2+ Doped Zn1-xCuxFe2O4 Spinel Nanoparticles using Williamson-Hall Plot Method

The nanoparticles (NPs) of Zn1-xCuxFe2O4 (ZCFO) spinels with x = 0, 0.2, 0.4, 0.6 and 0.8 were synthesized by a sol-gel combustion method using acetate precursor. The NPs of ZCFO were prepared by following calcination process at 600C for 8hrs. The synthesized NPs of ZCFO were characterized by X-ray diffraction (XRD) analysis using Rietveld refinement. The Rietveld refinement of the XRD patterns revealed that the ZCFO spinels crystallize into single diamond cubic structure with Fd-3m space group. The lattice constant and unit cell volume for ZCFO NPs shrink with enhancing doping concentration of Cu2+ ion. The crystalline growth in the NPs of ZCFO was examined by peak broadening present in the XRD pattern. The Williamson-Hall (W-H) plot method were used to study the individual involvements of crystallite sizes and lattice strain on the peak broadening of the NPs of ZCFO spinels. Whereas, particle size of the ZCFO sample with x = 0.40 was estimated by high-resolution scanning electron microscopy micrograph