Laboratory of Chemometrics, National Institute of Chemistry, Hajdrihova 19, Ljubljana, Slovenia.
J Comput Chem. 2013 Nov 5;34(29):2514-23. doi: 10.1002/jcc.23413. Epub 2013 Aug 19.
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.
对于非循环系统，已经知道图的中心要么是单个顶点，要么是两个相邻顶点，也就是边。如何将图中心的概念扩展到多环系统还不太清楚。文献中已经提出了几种多环分子图的图中心的方法。然而，在大多数情况下，尽管替代方法显然同样合理，但对于许多分子，它们给出了相同的结果，只是偶尔在分子中心的描述上有所不同。为了减少有资格成为图形中心的顶点数量，在寻找图形中心时已经考虑了规则的层次结构。我们通过使用图论的新概念——顶点“权重”，通过计算考虑的顶点与同一距离的顶点对的数量，重新考虑了“图形中心”的问题。这种方法对于非循环图的图中心与基于顶点偏心度的图中心的标准定义通常给出相同的结果。然而，在某些情况下，当找到两个非等价的顶点作为图中心时，新方法可以在两者之间进行区分。对于没有额外规则来定位形成多环图中心的顶点或顶点的循环图，同样的方法适用，这些顶点被称为图的中心顶点。此外，对于非循环、循环和多环图，新的顶点“权重”可以解释为顶点中心度，这是衡量顶点与图的中心或中心顶点的接近程度或距离的指标。除了说明一些较小的多环图的中心度之外，我们还报告了几个非循环图，它们的顶点具有相同的中心度值。
