The Extremal Graphs On The Zeroth-order General Randi(?) Index In Some Classes Of Graphs

Let G =(V,E) be a simple and connected graph with the vertex set V(G) and the edge set E(G),丨V(G)丨= n,丨E(G)丨= m be the number of vertex and edge of G,respectively.The zeroth-order general Randie index of the graph G is defined as P_o^a(G)=(?)d_v^a,where dv is the degree of vertex v,a is an arbitrary real number.The zeroth-order general Randie index of the graph is one of the most important topological indices in chemical graph theory.It has a lot of applications in chemistry, it has been widely investigated as well.Let G(n,n+1) be the set of simple connected bicyclic graphs with n vertices and n+1 edges,T_n,d the set of trees of order n and diameter d,C(n,k) the set of all connected cactuses with n vertices and k cycles.The zeroth-order general Randid index of the three classes of graphs above are investigated by using the graph sequence and graph transformations in this paper.The maximal and minimal zeroth-order general Randie index of graphs in G(n,n+1) are entirely characterized. We get the maximal zeroth-order general Randie index when a＞1 or a＜0,the minimal zeroth-order general Randie index when 0＜a<1 for the graphs in T_n,d and C(n,k),respectively, and the graphs with extremal values of the zeroth-order general Randie index are characterized.