Some mathematical properties of the geometric--arithmetic index/coindex of graphs

Abstract

Let $G=(V,E)$, $V=\{1,2,\ldots,n\}$, be a simple connected graph
of order $n$, size $m$ with vertex degree sequence $d_1\ge d_2\ge
\cdots \ge d_n> 0$, $d_i=d(v_i)$. The geometric--arithmetic topological index of $G$ is defined
as $GA(G)=\sum_{i \sim j}\frac{2\sqrt{d_id_j}}{d_i+d_j}$, whereas
the geometric--arithmetic coindex as
$\overline{GA}(G)=\sum_{i\nsim j} \frac{2\sqrt{d_id_j}}{d_i+d_j}$.
New lower bounds for $GA(G)$ and $\overline{GA}(G)$ in terms of
some graph parameters and other invariants are obtained.