On The Maximal General Abc Index Of Graphs With Fixed Maximum Degree

As a generalization of the famous atom-bond connectivity index ABC(G), the general atom-bound connectivity index of a graph G, ABC(alpha)(G), is denoted byABC(alpha)(G) = Sigma(uv is an element of E(G)) (d(G)(u) + d(G)(v) -2/d(G)(u)d(G)(v))(alpha) for any alpha is an element of R \ {0}.The (general) atom-bound connectivity index has been shown to be a useful topological index and has received more and more attention recently. In this paper, we show that ABC(alpha)(G + uv) > ABC(alpha)(G) holds for any two non-adjacent vertices u and v of a graph G with d(G)(u) + d(G)(v) >= 1 for 0 < alpha <= 1/2. Moreover, by applying this new property, we determine the maximum value of ABC(alpha) together with the corresponding extremal graphs in the class of graphs with n vertices and maximum degree Delta for 0 < alpha <= 1/2.