A note on the general zeroth-order Randic coindex of graphs

Let G be a simple graph of minimum degree at least 1. Let V = {v(1), v(2), v(n)} be the vertex set of G , and denote by d(i) the degree of the vertex v(i) for i = 1, 2, ... , n. If the tw o vertices v(i) and v(j) are not adjacent in G, we write i (sic) j. The general zeroth-order Randic coindex of G is defined as (R-0(alpha)) over bar (G) = Sigma(i(sic)j, i not equal j)(d(i)(alpha-1) + d(j)(alpha-1)), where alpha is an arbitrary real number. Denote by (G) over bar the complement of G . In this note, by assuming that G is a tree, we derive new lower bounds on the numbers (R-0(alpha)) over bar (G) and (R-0(alpha)) over bar (G), and determine all the graphs attaining these bounds. As the special al cases of the main results, we obtain bounds on the first Zagreb coindex (R-0(2)) over bar as well as on the forgotten topological coindex (R-0(3)) over bar (which is also called the Lanzhou index).