  Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a
vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$,
respectively. This paper proposes to investigate (both from mathematical and
applications points of view) those graph invariants of the form $\sum_{uv\in
E(G)}\varphi(d_v,d_w)$ in which $\varphi$ can be defined either using
well-known means of $d_v$ and $d_w$ (for example: arithmetic, geometric,
harmonic, quadratic, and cubic means) or by applying a basic arithmetic
operation (addition, subtraction, multiplication, and division) on any of two
such means, provided that $\varphi$ is a non-negative and symmetric function
defined on the Cartesian square of $D(G)$. Many existing well-known graph
invariants can be defined in this way; however, there are many exceptions too.
One of such uninvestigated graph invariants is the harmonic-arithmetic (HA)
index, which is obtained from the aforementioned setting by taking $\varphi$ as
the ratio of the harmonic and arithmetic means of $d_v$ and $d_w$. A molecular
tree is a tree whose maximum degree does not exceed four. Given the class of
all (molecular) trees with a fixed order, graphs that have the largest or least
value of the HA index are completely characterized in this paper.
