Location and multiplicities of eigenvalues for a star graph of Stieltjes strings

Abstract

The equations of motion of a star graph of Stieltjes strings with prescribed number of masses on each edge, with or without a mass at the central vertex, lead to a system of second order difference equations. At the central vertex Dirichlet or Neumann conditions are imposed while all pendant vertices are subject to Dirichlet conditions. We establish necessary and sufficient conditions on the location and multiplicities of two (finite) sequences of numbers and to be the corresponding Dirichlet and Neumann eigenvalues. Moreover, we derive necessary and sufficient conditions for one (finite) sequence to be the Neumann eigenvalues of such a star graph. Here the possible multiplicities play a key role; the conditions on them are formulated by means of the notion of vector majorization. Our results include, as a special case, some earlier results for star-patterned matrix inverse problems where only multiplicities, not the location of eigenvalues, are prescribed.