Densities in graphs and matroids

Certain graphs can be described by the distribution of the edges in its subgraphs. For example, a cycle C is a graph that satisfies |E(H)| |V (H)| < |E(C)| |V (C)| = 1 for all non-trivial subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)| |V (H)|−1 ≤ |E(T)| |V (T)|−1 = 1 for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such that |E(H)| |V (H)| ≤ |E(G)| |V (G)| and a 1-balanced graph is a graph such that |E(H)| |V (H)|−1 ≤ |E(G)| |V (G)|−1 for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G that satisfy the property |E(H)| ≤ k|V (H)| − l for all non-trivial subgraphs H of G play important roles in defining rigid structures. This dissertation is a formal study of a class of density functions that extends the above mentioned ideas. For a rational number r ≤ 1, a graph G is said to be r-balanced if and only if for each non-trivial subgraph H of G, we have |E(H)| |V (H)|−r ≤ |E(G)| |V (G)|−r . For r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined and the existence of these graphs is discussed. We also define a class of vulnerability measures on graphs similar to the edge-connectivity of graphs and show how it is related to r-balanced graphs. All these definitions are matroidal and the definitions of r-balanced matroids naturally extend the definitions of r-balanced graphs. The vulnerability measures in graphs that we define are ranked and are lesser than the edge-connectivity. Due to the relationship of the r-balanced graphs with the vulnerability measures defined in the dissertation, identifying r-balanced graphs and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced matroids and their weak forms are discussed. For r ∈ {0, 1}, we give a method to construct big r-balanced graphs from small r-balanced graphs. This construction is a generalization of the construction of Cartesian product of two graphs. We present an algorithmic solution of the problem of transforming any given graph into a 1-balanced graph on the same number of vertices and edges as the given graph. This result is extended to a density function defined on the power set of any set E via a pair of matroid rank functions defined on the power set of E. Many interesting results may be derived in the future by choosing suitable pairs of matroid rank functions and applying the above result

Balanced and 1-balanced graph constructions

AbstractThere are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced

N-acetyltransferase gene polymorphisms & plasma isoniazid concentrations in patients with tuberculosis

Background & objectives: Variations in the N-acetyltransferase (NAT2) gene among different populations could affect the metabolism and disposition of isoniazid (INH). This study was performed to genotype NAT2 gene polymorphisms in tuberculosis (TB) patients from Chennai, India, and compare plasma INH concentrations among the different genotypes. Methods: Adult patients with TB treated in the Revised National TB Control Programme (RNTCP) in Chennai, Tamil Nadu, were genotyped for NAT2 gene polymorphism, and two-hour post-dosing INH concentrations were compared between the different genotypes. Plasma INH was determined by high-performance liquid chromatography. Genotyping of the NAT2 gene polymorphism was performed by real-time polymerase chain reaction method. Results: Among the 326 patients genotyped, there were 189 (58%), 114 (35%) and 23 (7%) slow, intermediate and fast acetylators, respectively. The median two-hour INH concentrations in slow, intermediate and fast acetylators were 10.2, 8.1 and 4.1 μg/ml, respectively. The differences in INH concentrations among the three genotypes were significant (P<0.001). Interpretation & conclusions: Genotyping of TB patients from south India for NAT2 gene polymorphism revealed that 58 per cent of the study population comprised slow acetylators. Two-hour INH concentrations differed significantly among the three genotypes

An influence of dew point temperature on the occurrence of Mycobacterium tuberculosis disease in Chennai, India.

Climate factors such as dew point temperature, relative humidity and atmospheric temperature may be crucial for the spread of tuberculosis. This study was conducted for the first time to investigate the relationship of climatic factors with TB occurrence in an Indian setting. Daily tuberculosis notification data during 2008-2015 were generated from the National Treatment Elimination Program, and analogous daily climatic data were obtained from the Regional Meteorological Centre at Chennai city, Tamil Nadu, India. The decomposition method was adopted to split the series into deterministic and non-deterministic components, such as seasonal, non-seasonal, trend and cyclical, and non-deterministic climate factors. A generalized linear model was used to assess the relation independently. TB disease progression from latent stage infection to active was supported by higher dew point temperature and moderate temperature. It had a significant association with TB progression in the summer and monsoon seasons. The relative humidity may be favored in the winter and post-monsoon. The water tiny dew droplets may support the TB bacterium to recuperate in the environment

Consensus on Molecular Subtypes of High-grade Serous Ovarian Carcinoma

Purpose: The majority of ovarian carcinomas are of high-grade serous histology, which is associated with poor prognosis. Surgery and chemotherapy are the mainstay of treatment, and molecular characterization is necessary to lead the way to targeted therapeutic options. To this end, various computational methods for gene expression-based subtyping of high-grade serous ovarian carcinoma (HGSOC) have been proposed, but their overlap and robustness remain unknown. Experimental Design: We assess three major subtype classifiers by meta-analysis of publicly available expression data, and assess statistical criteria of subtype robustness and classifier concordance. We develop a consensus classifier that represents the subtype classifications of tumors based on the consensus of multiple methods, and outputs a confidence score. Using our compendium of expression data, we examine the possibility that a subset of tumors are unclassifiable based on currently proposed subtypes. Results: HGSOC subtyping classifiers exhibit moderate pairwise concordance across our data compendium (58.9%-70.9%, p \u3c 10-5) and are associated with overall survival in a metaanalysis across datasets (p \u3c 10-5). Current subtypes do not meet statistical criteria for robustness to re-clustering across multiple datasets (Prediction Strength \u3c 0.6). A new subtype classifier is trained on concordantly classified samples to yield a consensus classification of patient tumors that correlates with patient age, survival, tumor purity, and lymphocyte infiltration. Conclusion: A new consensus ovarian subtype classifier represents the consensus of methods, and demonstrates the importance of classification approaches for cancer that do not require all tumors to be assigned to a distinct subtype

Maximum Parsimony on Phylogenetic networks

Abstract Background Phylogenetic networks are generalizations of phylogenetic trees, that are used to model evolutionary events in various contexts. Several different methods and criteria have been introduced for reconstructing phylogenetic trees. Maximum Parsimony is a character-based approach that infers a phylogenetic tree by minimizing the total number of evolutionary steps required to explain a given set of data assigned on the leaves. Exact solutions for optimizing parsimony scores on phylogenetic trees have been introduced in the past. Results In this paper, we define the parsimony score on networks as the sum of the substitution costs along all the edges of the network; and show that certain well-known algorithms that calculate the optimum parsimony score on trees, such as Sankoff and Fitch algorithms extend naturally for networks, barring conflicting assignments at the reticulate vertices. We provide heuristics for finding the optimum parsimony scores on networks. Our algorithms can be applied for any cost matrix that may contain unequal substitution costs of transforming between different characters along different edges of the network. We analyzed this for experimental data on 10 leaves or fewer with at most 2 reticulations and found that for almost all networks, the bounds returned by the heuristics matched with the exhaustively determined optimum parsimony scores. Conclusion The parsimony score we define here does not directly reflect the cost of the best tree in the network that displays the evolution of the character. However, when searching for the most parsimonious network that describes a collection of characters, it becomes necessary to add additional cost considerations to prefer simpler structures, such as trees over networks. The parsimony score on a network that we describe here takes into account the substitution costs along the additional edges incident on each reticulate vertex, in addition to the substitution costs along the other edges which are common to all the branching patterns introduced by the reticulate vertices. Thus the score contains an in-built cost for the number of reticulate vertices in the network, and would provide a criterion that is comparable among all networks. Although the problem of finding the parsimony score on the network is believed to be computationally hard to solve, heuristics such as the ones described here would be beneficial in our efforts to find a most parsimonious network.</p