On Approximating the Number of kk-cliques in Sublinear Time

We study the problem of approximating the number of $k$-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let $n$ denote the number of vertices in the graph, $m$ the number of edges, and $C_k$ the number of $k$-cliques. We design an algorithm that outputs a $(1+\varepsilon)$-approximation (with high probability) for $C_k$, whose expected query complexity and running time are O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log n,1/\varepsilon,k). Hence, the complexity of the algorithm is sublinear in the size of the graph for $C_k = \omega(m^{k/2-1})$. Furthermore, we prove a lower bound showing that the query complexity of our algorithm is essentially optimal (up to the dependence on $\log n$, $1/\varepsilon$ and $k$). The previous results in this vein are by Feige (SICOMP 06) and by Goldreich and Ron (RSA 08) for edge counting ($k=2$) and by Eden et al. (FOCS 2015) for triangle counting ($k=3$). Our result matches the complexities of these results. The previous result by Eden et al. hinges on a certain amortization technique that works only for triangle counting, and does not generalize for larger cliques. We obtain a general algorithm that works for any $k\geq 3$ by designing a procedure that samples each $k$-clique incident to a given set $S$ of vertices with approximately equal probability. The primary difficulty is in finding cliques incident to purely high-degree vertices, since random sampling within neighbors has a low success probability. This is achieved by an algorithm that samples uniform random high degree vertices and a careful tradeoff between estimating cliques incident purely to high-degree vertices and those that include a low-degree vertex

Adhesion and proliferation of living cell on surface functionalized with glycine nanostructures

This research presents the application of glycine amino acid for establishing firm cell-substrate interaction instead of expensive adhesion proteins, peptides and peptide derivatives. The glycine amino acid is chemically functionalized on the coverslip to achieve self-assembled nanostructure. Glycine self-assembly on NaCl treated coverslips is initiated with SiONa+:COO− linkage while their nanostructure is achieved with formation of glycine chain through NH3+:COO− covalent linkage between the adjacent molecules. The functionalization steps are confirmed by Fourier-transform infrared spectroscopy (FTIR) investigation. The atomic force microscopy (AFM) and scanning electron microscopy (SEM) investigations reveal that glycine growth initiates at 4 Hours (H) post-treatment while maximum growth appears after 8H-10H. Both the vertical and horizontal growth of nanostructures show dependence on functionalization periods. Various levels of glycine functionalized surface show different levels of baby hamster kidney (BHK-21) cell adhesion and proliferation efficiency with maximum performance for 10H functionalized surface. The adhesion and proliferation performance of 10H glycine functionalized surface shows negligible difference when compared with glycine-aspartic acid (RGD) functionalized surface. Finally, growth curves obtained from both glycine and RGD functionalized surface reveal exponential growth phage up to 48H followed by stationary phage between 48H and 72H while death of many cells appears from 72H to 96H. Thus, this research concluded that glycine functionalized surface is equally effective for cell adhesion and proliferation

Association between human papillomavirus and endometrial adenocarcinoma

Several studies have suggested a possible role of human papillomavirus (HPV) in the pathogenesis of endometrial carcinoma. The aim of the study was to investigate the presence of HPV DNA in endometrium cancers and nonneoplastic endometrium. Sixty endometrial adenocarcinomas with and without squamous differentiation and the nonneoplastic endometrium tissue of fifty-six of the same patients were analyzed for the presence of family 16 and family 6 HPV DNA by using chromogenic in situ hybridization technique on formalin-fixed and paraffin-embedded archival samples, and the results were confirmed by polymerase chain reaction method. HPV DNA was not detected either in the endometrial adenocarcinoma with or without squamous differentiation, or in the nonneoplastic endometrium tissue. It appears that HPV does not play any role in the pathogenesis of endometrial carcinoma, since endometrium may not to be a suitable host for HPV replication. © Springer Science+Business Media New York 2013