Extremal properties of eigenvalues for a metric graph.
This paper deals with optimal lower bounds for the eigenvalues of the Laplace operator on a metric graph. Let \(\Gamma=(V,E)\) be a connected finite graph, where \(V\) is the set of its vertices, and \(E\) is the set of its edges. Each edge \(e\) is considered as a segment of certain length \(l(e)>0\) and call \(G=(\Gamma, (l(e)))\) a metric graph. The length of the metric graph \(G\) is defined as the sum of the length of all edges of \(G\). Let \(H_{k}\) stand for the star with \(k\) edges with equal length. For \(j\in{\mathbb{N}}\), let \(\lambda_{j}(G)\) be the \(j\)th eigenvalue of the Laplace operator on \(G\) subject to the Kirchhoff conditions on the vertices of \(G\). It is assumed that there are no vertices of degree 2 in \(G\). The author proves that \[ \mu_{j}(G)\geq\frac{\pi^{2}j^{2}}{4l(G)}\quad\text{for}\quad j\geq 2 \] and the equality holds if and only if \(G\) is a segment when \(j=2\) and \(G=H_{j}\) when \(j\geq 3\).
scientific article; zbMATH DE number 2162470
scientific article