Analysis of Eigenvalues for Molecular Structures

In this article,we study different molecular structures such as Polythiophene network,PLY(n)for n=1,2,and 3,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 for their cardinalities,chromatic numbers,graph variations,eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials.We convert the general structures of these chemical networks in to mathematical graphical structures.We transform the molecular structures of these chemical networks which are mentioned above,into a simple and undirected planar graph and sketch them with various techniques of mathematics.The matrices obtained from these simple undirected graphs are symmetric.We also label the molecular structures by assigning different colors.Their graphs have also been studied for analysis.For a connected graph,the eigenvalue that shows its peak point(largest value)obtained from the adjacency matrix has multiplicity 1.Therefore,the gap between the largest and its smallest eigenvalues is interlinked with some form of“connectivity measurement of the structural graph”.We also note that the chemical structures,Orthosilicate(Nesosilicate)SiO4,Pyrosilicates(Sorosilicates)Si2O7,Chain silicates(Pyroxenes)(SiO3)n,and Cyclic silicates(Ring Silicates)Si3O9 generally have two same eigenvalues.Adjacency matrices have great importance in the field of computer science.

Dynamic graph exploration by interactively linked node-link diagrams and matrix visualizations

The visualization of dynamic graphs is a challenging task owing to the various properties of the underlying relational data and the additional time-varying property.For sparse and small graphs,the most efficient approach to such visualization is node-link diagrams,whereas for dense graphs with attached data,adjacency matrices might be the better choice.Because graphs can contain both properties,being globally sparse and locally dense,a combination of several visual metaphors as well as static and dynamic visualizations is beneficial.In this paper,a visually and algorithmically scalable approach that provides views and perspectives on graphs as interactively linked node-link and adjacency matrix visualizations is described.As the novelty of this technique,insights such as clusters or anomalies from one or several combined views can be used to influence the layout or reordering of the other views.Moreover,the importance of nodes and node groups can be detected,computed,and visualized by considering several layout and reordering properties in combination as well as different edge properties for the same set of nodes.As an additional feature set,an automatic identification of groups,clusters,and outliers is provided over time,and based on the visual outcome of the node-link and matrix visualizations,the repertoire of the supported layout and matrix reordering techniques is extended,and more interaction techniques are provided when considering the dynamics of the graph data.Finally,a small user experiment was conducted to investigate the usability of the proposed approach.The usefulness of the proposed tool is illustrated by applying it to a graph dataset,such as e co-authorships,co-citations,and a Comprehensible Perl Archive Network distribution.