×

On the variance of topological indices that depend on the degree of a vertex. (English) Zbl 1274.05078

Summary: We present results on the variance of topological indices \(\mathcal{I}\) that are sums of \(f(\deg(u),\deg(v))\). where \(u,v\) are adjacent vertices and \(f\) is a function. The connectivity index or the 2nd Zagreb index are examples for indices of this kind. For random graphs on \(n\) vertices, we show that \(Var(\mathcal I)\) increases linearly in the number of vertices while \(Var(1/\sqrt n\mathcal I)\) remains bounded. Experiments with chemical structures and random graphs confirm our results. With a bounded variance, a better separation of size-dependent and size-independent properties is obtained. The results are important for the processing of descriptor data with neural nets.

MSC:

05C07 Vertex degrees
05C10 Planar graphs; geometric and topological aspects of graph theory
05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDFBibTeX XMLCite