Hamiltonian Spectra of Trees

Abstract

Let $G$ be a connected graph, and let $d(u,v)$ denote the distance between vertices $u$ and $v$ in $G$. For any cyclic ordering $\pi$ of $V(G)$, let $\pi =(v_1,v_2,\dots ,v_n,v_{n+1}=v_1)$, and let $d(\pi )=\sum _{i=1}^{n}d(v_i,v_{i+1})$. The set of possible values of $d(\pi )$ of all cyclic orderings $\pi$ of $V(G)$ is called the Hamiltonian spectrum of $G$. We determine the Hamiltonian spectrum for any tree.