Criteria for the Existence of Systems of Subspaces Related to a Certain Class of Unicyclic Graphs

We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length m ≥ 3 with chains of length s ≥ 1 attached to each vertex of the cycle. There is a one-to-one correspondence between the vertices of the graph and the analyzed subspaces. If an edge connects two vertices, then the angle between the subspaces is equal to 𝜓 𝜖 (0; 𝜋/2); otherwise, the subspaces are orthogonal. Applying the theorem on reduction of unicyclic graph, we prove that nontrivial configurations exist if and only if cos 𝜓 𝜖 (0; 𝜏_m,s]. We also deduced formulas for 𝜏_m,s and showed that $\bigcup_{m,s}\left(\left.0;{\tau}_{m,s}\right]=\right.\left(\left.0;2/5\right]\right.$.