We consider the problem of leaky waves in an inhomogeneous waveguide structure coveredwith a layer of graphene, which is reduced to a boundary value problem for the longitudinalcomponents of the electromagnetic field in Sobolev spaces. A variational statement of the problemis used to determine the solution. The variational problem is reduced to the study of an operatorfunction. The properties of the operator function necessary for the analysis of its spectralproperties are investigated. Theorems on the discreteness of the spectrum and on the distributionof the characteristic numbers of the operator function on the complex plane are proved.