The cluster-concentrated eigenvector basis conjecture for graph Paley–Wiener spaces

Let $0<K<N$. Let $P$ be the orthogonal projection onto $\operatorname{PW}_{2K}(\mathcal{B}_N\vdash\mathcal{C}_m)$, and let $Q$ be the projection defined above. Write $\operatorname{dim}_K$ for the dimension parameter used in the construction. The cyclic shift acts on $\ell^2(\mathcal{B}_N\vdash\mathcal{C}_m)$.

Cluster-concentrated basis conjecture. There are $\operatorname{dim}_K-1$ eigenvalues of $PQ$ larger than $1/2$, including $\operatorname{dim}_K-(K+1)$ eigenvalues equal to one and another $K$ eigenvalues in $(1/2,1)$. The cyclic shifts of the corresponding eigenvectors are linearly independent in $\ell^2(\mathcal{B}_N\vdash\mathcal{C}_m)$. Therefore, these vectors form a basis for $\operatorname{PW}_{2K}$, which has dimension $m(\operatorname{dim}_K-1)$.

This claim describes the proposed measurement-vector construction for sampling low-spectrum signals on the graph formed from the bouquet and cycle. The supplied passage gives the eigenvector and projection framework but does not provide evidence that the conjecture has been proved or disproved.