Non-canonical Connectivity Measures on Metric Graphs

Explore the concept of non-canonical connectivity measures on metric graphs in this seminar talk from the Spectral Geometry in the clouds series. Delve into Delio Mugnolo's presentation, which examines alternatives to Fiedler's "algebraic connectivity" for measuring graph connectivity. Learn about the spectral gap of the metric graph Laplacian and its relationship to connectivity, as established by Nicaise and Kennedy-Kurasov-Malenová-Mugnolo. Investigate the mean distance as a geometric measure of connectivity in compact metric measure spaces, and discover its interplay with the spectral gap of the metric graph Laplacian. Gain insights into the similarities between this approach and the Kohler-Jobin inequality for metric graphs. This talk, based on joint work with Luis Baptista and James B. Kennedy, offers a deep dive into advanced mathematical concepts at the intersection of spectral geometry and graph theory.