On the spectral gap of a quantum graph
The main goal of the paper is to initiate a systematic study of the spectral geometry of quantum graphs. As a starting point, the authors consider connected, compact and finite metric graphs equipped with a Neumann (Kirchoff) Laplacian. They investigate the possibility to obtain, eventually sharp, lower and upper estimates of the spectral gap (the first positive eigenvalue, in this case) in terms of the total length of the graph \(L\), of its diameter \(D\), of the number of vertices \(V\) or of the number of its edges \(E\). One gives or obtains: lower estimates in terms of \(L\); upper estimates in terms of \(L\) and \(E\); upper estimates in terms of \(D\) and \(V\); upper and lower estimates in terms of \(D\) and \(L\). Using extremal graphs, one shows that some of these estimates are sharp. The authors also prove that it is not possible to obtain universal estimates in terms of \(D\) only, in contrast to the Laplacian matrix on a combinatorial graph. The proof of this assertion is done using pumpkin graphs and a reduction to a Sturm-Liuville problem. In the last section, estimates for the normalised Laplacian matrix on combinatorial graphs are deduced. A final remark: it is a pleasure to read such a paper.
scientific article; zbMATH DE number 6627409
scientific article