Maximum zeroth-order general Randić index of orientations of cacti

Abstract

The zeroth-order general Randić index $R_{a+1}^0$ of an $n$-vertices oriented graph $D$ is equal to the sum of $(d_{u_i}^{+}{)}^a+(d_{u_j}^{-}{)}^a$ over all arcs $u_i{u}_j$ of $D$, where we denote by $d_{u_i}^{+}$ the out-degree of the vertex $u_i$ and $d_{u_j}^{-}$ the in-degree of the vertex $u_j$, $a$ is an arbitrary real number. In the paper, we determine the orientations of cacti with the maximum value of the zeroth-order general Randić index for $a\ge 1$.

zeroth-order general Randić index; cactus; extremal graph