An analog of the Jordan–Dirichlet theorem for an operator with involution on a graph

In this paper, we examine the convergence of eigenfunction expansions of a functional-differential operator with involution $\nu(x)=1-x$, which is defined on a geometric graph consisting of two edges, one of which is a loop. Sufficient conditions are obtained for the uniform convergence of the Fourier series in the eigenfunctions of the operator (an analog of the Jordan–Dirichlet theorem).