  A graph $G$ on $n$ vertices of diameter $D$ is called $H$-palindromic if
$\alpha(G,k) = \alpha(G,D-k)$ for all $k=0, 1, \dots, \left
\lfloor{\frac{D}{2}}\right \rfloor$, where $\alpha(G,k)$ is the number of
unordered pairs of vertices at distance $k$. Quantities $\alpha(G,k)$ form
coefficients of the Hosoya polynomial. In 1999, Caporossi, Dobrynin, Gutman and
Hansen showed that there are exactly five $H$-palindromic trees of even
diameter and conjectured that there are no such trees of odd diameter. We prove
this conjecture for bipartite graphs. An infinite family of $H$-palindromic
trees of diameter $6$ is also constructed.
