Let G=(V,E) be a simple connected graph with vertex set V(G) and edge set
E(G). The third atom-bond connectivity index, ABC3β index, of G is
defined as ABC3β(G)=uvβE(G)ββe(u)e(v)e(u)+e(v)β2ββ, where eccentricity e(u) is the
largest distance between u and any other vertex of G, namely
e(u)=max{d(u,v)β£vβV(G)}. This work determines the maximal ABC3β index
of unicyclic graphs with any given girth and trees with any given diameter, and
characterizes the corresponding graphs