Marija Rasajski

Fitting a geometric graph to a protein–protein interaction network

MOTIVATIONnFinding a good network null model for protein-protein interaction (PPI) networks is a fundamental issue. Such a model would provide insights into the interplay between network structure and biological function as well as into evolution. Also, network (graph) models are used to guide biological experiments and discover new biological features. It has been proposed that geometric random graphs are a good model for PPI networks. In a geometric random graph, nodes correspond to uniformly randomly distributed points in a metric space and edges (links) exist between pairs of nodes for which the corresponding points in the metric space are close enough according to some distance norm. Computational experiments have revealed close matches between key topological properties of PPI networks and geometric random graph models. In this work, we push the comparison further by exploiting the fact that the geometric property can be tested for directly. To this end, we develop an algorithm that takes PPI interaction data and embeds proteins into a low-dimensional Euclidean space, under the premise that connectivity information corresponds to Euclidean proximity, as in geometric-random graphs. We judge the sensitivity and specificity of the fit by computing the area under the Receiver Operator Characteristic (ROC) curve. The network embedding algorithm is based on multi-dimensional scaling, with the square root of the path length in a network playing the role of the Euclidean distance in the Euclidean space. The algorithm exploits sparsity for computational efficiency, and requires only a few sparse matrix multiplications, giving a complexity of O(N(2)) where N is the number of proteins.nnnRESULTSnThe algorithm has been verified in the sense that it successfully rediscovers the geometric structure in artificially constructed geometric networks, even when noise is added by re-wiring some links. Applying the algorithm to 19 publicly available PPI networks of various organisms indicated that: (a) geometric effects are present and (b) two-dimensional Euclidean space is generally as effective as higher dimensional Euclidean space for explaining the connectivity. Testing on a high-confidence yeast data set produced a very strong indication of geometric structure (area under the ROC curve of 0.89), with this network being essentially indistinguishable from a noisy geometric network. Overall, the results add support to the hypothesis that PPI networks have a geometric structure.nnnAVAILABILITYnMATLAB code implementing the algorithm is available upon request.

Multicyclic treelike reflexive graphs

A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles.

On bicyclic reflexive graphs

A simple graph is said to be reflexive if the second largest eigenvalue of a (0,1)-adjacency matrix does not exceed 2. We use graph modifications involving Smith trees to construct four classes of maximal bicyclic reflexive graphs.

Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities

In this paper we propose a new method for sharpening and refinements of some trigonometric inequalities. We apply these ideas to some inequalities of Wilker–Cusa–Huygens type.

Refined Estimates and Generalizations of Inequalities Related to the Arctangent Function and Shafer’s Inequality

In this paper we give some sharper refinements and generalizations of inequalities related to Shafers inequality for the arctangent function, stated in Theorems 1, 2 and 4 in [1], by C. Mortici and H.M. Srivastava.

Decomposition of Smith graphs in maximal reflexive cacti

The spectrum of a graph is the family of eigenvalues of its (0,1) adjacency matrix. A simple graph is reflexive if its second largest eigenvalue @l2 does not exceed 2. The graphic property @l2=<2 is a hereditary one, i.e. every induced subgraph of a reflexive graph preserves this property and that is why reflexive graphs are usually represented through maximal graphs. Cacti, or treelike graphs, are graphs whose all cycles are mutually edge-disjoint. The set of simple connected graphs characterized by the property @l1=2, where @l1 is the largest eigenvalue, is known as the set of Smith graphs. It consists of cycles of all possible lengths and some trees. If two trees T1 and T2 have such vertices u1@?T1 and u2@?T2 which, after their identification u1=u2=u give a Smith tree, we say that that Smith tree can be split at its vertex u into T1 and T2. It has turned out that several classes of maximal reflexive cacti can be described in the following way: we start from certain essential cyclic structure with two characteristic vertices c1 and c2, and then form a family of maximal connected reflexive cacti by splitting Smith trees, and by attaching their parts to c1 and c2. This way of decomposition of Smith trees leads to an interesting phenomenon of so-called pouring of Smith trees between two vertices.

Refinements and generalizations of some inequalities of Shafer-Fink’s type for the inverse sine function

In this paper, we give some sharper refinements and generalizations of inequalities related to Shafer-Fink’s inequality for the inverse sine function stated in Theorems 1, 2, and 3 of Bercu (Math. Probl. Eng. 2017: Article ID 9237932, 2017).

Sharpening and generalizations of Shafer-Fink and Wilker type inequalities: a new approach

In this paper we propose and prove some generalizations and sharpenings of certain inequalities of Wilker;s and Shafer-Finks type. Application of the Wu-Debnath theorem enabled us to prove some double sided inequalities.

A New Method for Proving Some Inequalities Related to Several Special Functions

In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential inequalities.

About some exponential inequalities related to the sinc function

In this paper, we prove some exponential inequalities involving the sinc function. We analyze and prove inequalities with constant exponents and inequalities with certain polynomial exponents. Also, we establish intervals in which these inequalities hold.