Upper bound for the diameter of a tree in the quantum graph theory
Abstract
UDC 519.177
We study two Sturm – Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at internal vertices and Neumann conditions at pendant vertices (first problem) and with Dirichlet conditions at pendant vertices (second problem). The spectrum of each of these problems consists of infinitely many normal (isolated Fredholm) eigenvalues. It is shown that, knowing the asymptotics of the eigenvalues, it is possible to estimate the diameter of a tree from above for each of these problems.
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