The general zeroth-order Randic index of maximal outerplanar graphs and trees with k maximum degree vertices

For a graph, the general zeroth-order Randic index R-alpha(0) is defined as the sum of the alpha th power of the vertex degrees (alpha not equal 0, alpha not equal 1). Let H-n be the class of all maximal outerplanar graphs on n vertices, and T-n,T-k be the class of trees with n vertices of which k vertices have the maximum degree. We first present a lower bound (respectively, upper bound) for the general zeroth-order Randic index of graphs in H-n (respectively, T-n,T-k) when alpha is an element of(-infinity, 0) boolean OR (1, + infinity) (respectively, alpha is an element of (2, + infinity)), and characterize the extremal graphs. Then we determine graphs of the class T-n,T-k with maximal and minimal general zeroth-order Randic index when alpha is an element of(-infinity, 0) boolean OR (1, + infinity), respectively.