Abstract
For any real number a, the sum of the alpha-th powers of the degrees of a (molecular) graph G, denoted by R-0(alpha)(G), is known as the general zeroth-order Rancho index as well as the general first Zagreb index and variable first Zagreb index. Research on the graph invariant R-0(alpha) (for specific values of alpha) began in the 1970s, when the first Zagreb index R-0(2) and the zeroth-order connectivity/Randie index R-0(-1/2) were introduced within the study of molecular modeling. After that, several other specific versions of the invariant R-0(alpha) were also studied. These versions include inverse degree (or modified total adjacency index) R-0(-1), modified first Zagreb index R-0(-2), and forgotten topological index R-0(3). The main purpose of the present survey is to present bounds and extremal results related to the invariant R-0(alpha), including all the aforementioned specific versions of R-0(alpha).