  Very recently, the first geometric-arithmetic index $GA$ and
arithmetic-geometric index $AG$ were introduced in mathematical chemistry. In
the present paper, we first obtain some lower and upper bounds on $AG$ and
characterize the extremal graphs. We also establish various relations between
$AG$ and other topological indices, such as the first geometric-arithmetic
index $GA$, atom-bond-connectivity index $ABC$, symmetric division deg index
$SDD$, chromatic number $\chi$ and so on. Finally, we present some sufficient
conditions of $GA(G)>GA(G-e)$ or $AG(G)>AG(G-e)$ for an edge $e$ of a graph
$G$. In particular, for the first geometric-arithmetic index, we also give a
refinement of Bollob\'{a}s-Erd\H{o}s-type theorem obtained in [3].
