Rank one density property for a class of $M$-bases

In the early 1990s the works of Larson, Wogen and Argyros, Lambrou, Longstaff disclosed an example of a strong $M$-basis that did not admit a linear summation method. We study a class of $M$-bases $\mathfrak{F}=\{f_n\}_{n=1}^\infty$ in the Hilbert space that generalizes the Larson--Wogen system. We determine the conditions under which $\mathfrak{F}$ admits a linear summation method. In order to do that we employ some of the graph theory techniques.