On the palindromic Hosoya polynomial of trees

TL;DR

This paper investigates the symmetry properties of the Hosoya polynomial in trees, proving a conjecture about the absence of odd-diameter $H$-palindromic trees in bipartite graphs and constructing an example of even-diameter trees.

Contribution

It proves the conjecture that no odd-diameter $H$-palindromic trees exist in bipartite graphs and constructs an infinite family of even-diameter $H$-palindromic trees.

Findings

No $H$-palindromic trees of odd diameter in bipartite graphs.

Existence of an infinite family of $H$-palindromic trees with diameter 6.

Confirmation of the conjecture for bipartite graphs.

Abstract

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.