On topological indices for small RNA graphs

The secondary structure of RNAs can be represented by graphs at various resolutions. While it was shown that RNA secondary structures can be represented by coarse grain tree-graphs and meaningful topological indices can be used to distinguish between various structures, small RNAs are needed to be represented by full graphs. No meaningful topological index has yet been suggested for the analysis of such type of RNA graphs. Recalling that the second eigenvalue of the Laplacian matrix can be used to track topological changes in the case of coarse grain tree-graphs, it is plausible to assume that a topological index such as the Wiener index that represents all Laplacian eigenvalues may provide a similar guide for full graphs. However, by its original definition, the Wiener index was defined for acyclic graphs. Nevertheless, similarly to cyclic chemical graphs, small RNA graphs can be analyzed using elementary cuts, which enables the calculation of topological indices for small RNAs in an intuitive way. We show how to calculate a structural descriptor that is suitable for cyclic graphs, the Szeged index, for small RNA graphs by elementary cuts. We discuss potential uses of such a procedure that considers all eigenvalues of the associated Laplacian matrices to quantify the topology of small RNA graphs.