On the construction and comparison of graph irregularity indices

Abstract

Irregularity indices are generally used for quantitative characterization of topological structure of non-regular graphs. According to a widely accepted preconception, using a topological invariant (called a graph irregularity index) for that purpose, the results of graph irregularity classification should be consistent with our subjective judgements (intuitive feeling). In the case of structurally strongly similar graphs, it is difficult to select the proper irregularity index by which the irregularity ranking (ordering) of graphs can be per- formed in a consistent manner to our preliminary expectations. In this work we investigate various possibilities of constructing so-called composite irregularity indices obtained as a function of traditional irregularity indices and test their discriminatory performance. More- over, it has been demonstrated in examples, that in some cases the subjective evaluation of graph irregularities leads to false conclusions. This phenomenon is called the paradox of quantitative graph irregularity characterization. From our study we have concluded that results of graph irregularity measuring depend not only on the choice of irregularity indices but it is inuenced strongly on the preselected set of graphs to be investigated. Similar problems arise for the quantitative evaluation of information content, complexity or branching of molecular graphs.