Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.Comment: 30 pages, 8 figure

Permutability graphs of subgroups of some finite non-abelian groups

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups $D_n$, the generalized quaternion groups $Q_n$, the quasi-dihedral groups $QD_{2^n}$ and the modular groups $M_{p^n}$. Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.Comment: 35 pages, 1 figur

Novel Molecules for Intra-Oral Delivery of Antimicrobials to Prevent and Treat Oral Infectious Diseases

New molecules were designed for efficient intra-oral delivery of antimicrobials to prevent and treat oral infection. The salivary statherin fragment, which has high affinity for the tooth enamel, was used as a carrier peptide. This was linked through the side chain of the N-terminal residue to the C-terminus of a defensin-like 12-residue peptide to generate two bifunctional hybrid molecules, one with an ester linkage and the other with an anhydride bond between the carrier and the antimicrobial components. They were examined for their affinity to a HAP (hydroxyapatite) surface. The extent of the antimicrobial release in human whole saliva was determined using 13C-NMR spectroscopy. The candidacidal activity of the molecules was determined as a function of the antimicrobial release from the carrier peptide in human saliva. The hybrid-adsorbed HAP surface was examined against Candida albicans and Aggregatibacter actinomycetemcomitans using the fluorescence technique. The bifunctional molecules were tested on human erythrocytes, GECs (gingival epithelial cells) and GFCs (gingival fibroblast cells) for cytotoxicity. They were found to possess high affinity for the HAP mineral. In human whole saliva, a sustained antimicrobial release over a period of more than 40–60 h, and candidacidal activity consistent with the extent of hybrid dissociation were observed. Moreover, the bifunctional peptide-bound HAP surface was found to exhibit antimicrobial activity when suspended in clarified human saliva. The hybrid peptides did not show any toxic influence on human erythrocytes, GECs and GFCs. These novel hybrids could be safely used to deliver therapeutic agents intra-orally for the treatment and prevention of oral infectious diseases

The complement of proper power graphs of finite groups

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes have been mad

Resistivity and Thermopower of Ni2.19Mn0.81Ga

In this paper, we report results of the first studies on the thermoelectric power (TEP) of the magnetic heusler alloy Ni$_{2.19}$Mn$_{0.81}$Ga. We explain the observed temperature dependence of the TEP in terms of the crystal field (CF) splitting and compare the observed behavior to that of the stoichiometric system Ni$_2$MnGa. The resistivity as a function of temperature of the two systems serves to define the structural transition temperature, T$_M$, which is the transition from the high temperature austenitic phase to low temperatures the martensitic phase. Occurrence of magnetic (Curie-Weiss) and the martensitic transition at almost the same temperature in Ni$_{2.19}$Mn$_{0.81}$Ga has been explained from TEP to be due to changes in the density of states (DOS) at the Fermi level.Comment: 12 pages, 4 figures, Accepted in Physical Review B vol 70, Issue 1