Common vertex matrix: A novel characterization of molecular graphs by counting

Randić, Milan; Novič, Marjana; Plavšić, Dejan

doi:10.1002/jcc.23300

Cited by 3 publications

(1 citation statement)

References 43 publications

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“…Recently, a novel graph matrix has been introduced in chemical graph theory, [18] the matrix element (i, j) of which counts the number of vertices at equal distance from vertices i and j. This matrix, referred to as "common vertex matrix" has a number of interesting properties, including high power of discrimination among graphs having the same number of vertices.…”

Section: Novel Characterization Of Graph Central Verticesmentioning

confidence: 99%

On the centrality of vertices of molecular graphs

et al. 2013

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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.

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Section: Novel Characterization Of Graph Central Verticesmentioning

confidence: 99%