  It is proved that the median eigenvalues of every connected bipartite graph
$G$ of maximum degree at most three belong to the interval $[-1,1]$ with a
single exception of the Heawood graph, whose median eigenvalues are
$\pm\sqrt{2}$. Moreover, if $G$ is not isomorphic to the Heawood graph, then a
positive fraction of its median eigenvalues lie in the interval $[-1,1]$. This
surprising result has been motivated by the problem about HOMO-LUMO separation
that arises in mathematical chemistry.
