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Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is astatement about the eigenvalues of the linear operator obtained from thequantum multiplication of the anticanonical class of $X$. Conjecture$\mathcal{O}$ is a conjecture that Property $\mathcal{O}$ holds for any Fanovariety. Pasquier listed the smooth non-homogeneous horospherical varieties ofPicard rank 1 into five classes. Conjecture $\mathcal{O}$ has already beenshown to hold for the odd symplectic Grassmannians which is one of theseclasses. We will show that Conjecture $\mathcal{O}$ holds for two more classesand an example in a third class of Pasquier's list. The theory ofPerron-Frobenius reduces our proofs to be graph-theoretic in nature.

Conjecture $\mathcal{O}$ holds for some Horospherical Varieties of Picard Rank 1