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arXiv · 2112.11164

On the palindromic Hosoya polynomial of trees

Dmitry Badulin·Alexandr Grebennikov·Konstantin Vorob'ev
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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.

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Dmitry Badulin, Alexandr Grebennikov, Konstantin Vorob'ev. 2021-12-21. On the palindromic Hosoya polynomial of trees. https://arxiv.org/abs/2112.11164

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