Several Characterizations on Degree-Based Topological Indices for Star of David Network

In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as breaking point, protection, and strain vitality, correspond to these TIs. In the compound graph hypothesis, the concept of TIs was developed in view of the degree of vertices. In investigating minimizing exercises of Star of David, these indices are useful. In this study, we explore the different types of Zagreb indices, Randić indices, atom-bond connectivity indices, redefined Zagreb indices, and geometric-arithmetic index for the Star of David. The edge partitions of this network are tabled based on the sum of degrees-of-end vertices and the sum of degree-based edges. To produce closed formulas for some degree-based network TIs, these edge partitions are employed.

Generalizing the Divisibility Property of Rectangle Domino Tilings

We introduce a class of graphs called compound graphs, which are constructed out of copies of a planar bipartite base graph, and explore the number of perfect matchings of compound graphs. The main result is that the number of matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's Enumeration of Matchings, a divisibility property of rectangles. Finally, we present a new proof, in the same spirit, of Ciucu's factorization theorem.

A New Constraint-Based Compound Graph Layout Algorithm for Drawing Biochemical Networks

Due to the huge amount of information available in biochemical databases, biologists need sophisticated tools to accurately extract the information from such databases and to interpret it correctly. Those tools must be able to dynamically generate any kind of biochemical subgraph (i.e., metabolic pathways, genetic regulation, signal transduction, etc.) in a single graph. The visualization tools must be able to cope with such graphs and to take into account the particular semantics of all kinds of biochemical subgraphs. Therefore, such tools need generic graph layout algorithms that adapt their behavior to the data semantics. In this paper we present the Constrained Compound Graph Layout (C2GL) algorithm designed for the generic representation of biochemical graphs and in which users can represent knowledge about how to draw graphs in accordance with the biochemical semantics. We show how we implemented the C2GL algorithm in the Visual BioMaze framework, the visualization tool of the BioMaze project.