Criteria for the existence of systems of subspaces related to a certain class of unicyclic graphs


UDC 512.552.4
We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length $m\geqslant 3$ and has, at each vertex of the cycle, a chains of length $s\geqslant 1$ glued to the vertex.
There is a one-to-one correspondence between the vertices and subspaces.
If an edge connects two vertices, then the angle between subspaces is equal to $\psi\in(0;\pi/2),$ otherwise the subspaces are orthogonal.
Applying the theorem on reduction of unicyclic graph, we prove that nonzero configurations exist if and only if $\cos\psi\in(0;\tau_{m,s}].$
We obtain formulas for $\tau_{m,s}$ and show that~$\bigcap\limits_{m,s}(0;\tau_{m,s}] = (0;2/5].$


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How to Cite
Popova, N. D., and O. V. Strilets. “ Criteria for the Existence of Systems of Subspaces Related to a Certain Class of Unicyclic Graphs”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 4, Apr. 2021, pp. 556 -5, doi:10.37863/umzh.v73i4.6354.
Research articles