An overview on Randic (Normalized Laplacian) spread

Abstract

The definition of Randic matrix comes from a molecular structure descriptor introduced by Milan Randic in 1975, known as Randic index. The plethora of chemical and pharmacological applications of the Randic index, as well as numerous mathematical investigations are well known and presented in the literature. In spite of its connection with Randic index this matrix seems to have not been much studied in mathematical chemistry however, some graph invariants related with this matrix such as Randic energy (the sum of the absolute values of the eigenvalues of the Randic matrix), the concept of Randic spread (that is, the maximum difference between two eigenvalues of the Randic matrix, disregarding the spectral radius) were recently introduced and some of their properties were established. We review some topics related with the graph invariant Randic spread, such as bounds that were obtained from matrix and/or numerical inequalities establishing relations between this spectral parameter and some structural parameters of the underlying graph. Moreover, some new bounds for the Randic spread are obtained. Comparisons with some upper bounds for the Randic spread of regular graphs are done. Finally, a possible relation between Randic spread and Randic energy is established.