On the centrality of vertices of molecular graphs

Abstract

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of "the center of a graph" by using a novel concept of graph theory, the vertex "weights," defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex "weights," in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices.

Differentiation of alkyl radicals: A route through chemical graph theory

“…Finally, the depiction of core structure of alkyl radicals in the form of degree matrix (DM) [42] in Figure 5 captures the important nodes of the T ‐junctions of the target radicals, C (1) and C (9) , respectively. This position C (1) with DM value of 3 indicates a three‐way connection at this atom and locates the unpaired electron of the radicals and therefore, the most important chemical spot of radicals.…”

Nanoinformatics in Drug Delivery

“…The elements of the matrix are constructed to show which pairs of atoms in the molecule are adjacent, and which are not. [14] Additionally, the thousands of molecular descriptors can all be mathematically weighted by different chemical properties such as the ionizing potential (Figure 1c-d), which adds another layer of information to the descriptors. There are different software programs that calculate descriptors for QSAR/QSPR either free of charge or with a purchased license (Table 1).…”

Milestones in graphical bioinformatics

“…It is obvious that many invariants are needed to capture salient features of information-rich proteomics maps. The use of partial ordering is one of several routes to the numerical characterization of proteomics maps, and represents a continuation of our efforts [81][82][83][84][85][86][87] to develop the mathematical characterization of DNA, proteins, and proteome. The partial ordering diagram shown in Figure 18 is based on protein spots ordered with respect to their charge and mass, and connecting lines are embedded over the proteomics map.…”

“…A way to arrive at a sparse matrix for a map is to consider geometrical objects that overlap the map. We illustrate (1) the construction of the partial ordering graph of the map vertices [105,106] ; and (2) the construction of the graph of sequential nearest neighbors for the vertices of the map. [107] In both cases, maps can be represented by a sparse binary matrix.…”