Keywords: exact dynamic stiffness matrix, Wittrick-Williams algorithm, infinite order determinant, graph theory, Sturm-Liouville operator.

Quantum graph problems occur in many disciplines of science and engineering and they can be solved by viewing the problem as a structural engineering one. The Sturm-Liouville operator acting on a tree is an example of a quantum graph and the structural engineering analogy is the axial vibration of an assembly of bars with identical topology. Previous work has demonstrated that a numerical procedure can be used to provide insight into the spectral behaviour of the free Laplacian in one dimension acting on a metric tree. In the case of an infinitely long tree, the eigenvalues form bands of continuous spectra with an eigenvalue of infinite multiplicity in the middle of the gaps. In this paper, as in previous work, only finite length trees are considered.

The required eigenvalues are found using the well-established Wittrick-Williams algorithm in conjunction with a dynamic stiffness approach. It is known that the plot of the determinant of the stiffness matrix versus a trial eigenparameter has several poles and that the eigenvalues of the system can occur when the determinant of the stiffness matrix is either zero or infinite. For a finite tree there are no bands of continuous spectra, however all eigenvalues fall within the lower and upper bounds of each band and the bands are repetitive.

The determinant plot starts from a positive value at zero on the abscissa and after crossing the abscissa one or more times tends to infinity as the trial eigenparameter tends to the clamped/clamped member eigenvalue. An established procedure transforms this plot into what is known as the 'infinite order determinant', in which there are no poles and all the original eigenvalues now correspond to the determinant being zero. These properties are used in this paper to show that although the transformed plot tends to zero for large values of trial eigenparameter, a small modification can produce a plot that repeats at regular intervals and corresponds to the band gap structure of the spectrum for an infinitely long tree. From the results of finite length trees it is then possible to estimate the edges of the spectrum for the infinite system. Furthermore, it is known that the eigenvalues of finite trees are obtainable from a family of stepped bars. This offers the possibility of significantly reducing the order of the problem to be solved and leads to a very efficient way of estimating the spectral limits.

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