On torsional rigidity and ground-state energy of compact quantum graphs

Abstract

We develop the theory of torsional rigidity—a quantity routinely considered for Dirichlet Laplacians on bounded planar domains—for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph’s vertices and is, thus, much easier than computing eigenvalues.

Torsional rigidity in random walk spaces

“…As a consequence of the above theorem and (6.9) we recover the equivalent to Proposition 4.8 of [35]. 2.…”

“…Torsional rigidity on Quantum Graphs as a m-torsional rigidity on graphs Torsional rigidity on quantum graphs was introduce by Colladay, Kaganovskiy and McDonald in [14]. To the best of our knowledge, after this paper, the only existing literature on this topic is the paper by Mugnolo and Plumer [35], where the torsional rigidity of a quantum graph is related to the rigidity of an associated weighted combinatorial graph. We will interpret here that result with the (nonlocal) rigidity of a weighted graph.…”

“…Then we have that formula (7) given in [35] can be written using weighted discrete graphs, seen as random walk spaces, as follows. whatever c is chosen in (7).…”

“…To the best of our knowledge most of the results we get are new even for the particular cases of locally finite weighted graphs and nonlocal problems in domains of R N . Finally we relate the torsional rigidity given here for graphs with the torsional rigidity on metric graphs stated in [35].…”

Pointwise eigenvector estimates by landscape functions: some variations on the Filoche--Mayboroda--van den Berg bound

“…Theorem 1.1 especially applies to large classes of (not necessarily self-adjoint) uniformly elliptic and even degenerate second order operators satisfying a maximum principle [48,Chapters 4 and 5], and in particular to Schrödinger operators with a suitably positive potential with Dirichlet or Neumann conditions on bounded open domains of R d or on suitable subset of compact manifolds -this is the setting discussed in [21,3] -but also on metric [31,46] or combinatorial graphs [23]: in these settings, again, ρ = 1 is the usual choice.…”