Root-radius bounds and stability for p-torsion on compact metric graphs
Year: 2026
Type: preprint
Abstract: Let a finite connected compact metric graph of total length L have one Dirichlet leaf r, with nonlinear Kirchhoff conditions at every other vertex. For 1 < p < infinity, put q = p/(p-1), let w_p be the p-torsion function, and set M_p = integral_G w_p. If R is the largest graph distance from r, we prove M_p <= (L^(q+1) - (L-R)^(q+1))/(q+1). Thus the normalized Saint-Venant deficit is at least (1-R/L)^(q+1). The coefficient 1 and the power q+1 are optimal over all finite graph topologies, including the small-deficit regime. The proof cuts cycles while preserving every point distance from the root and uses a nonnegative deficit decomposition on the resulting tree. The results include a long geodesic spine, rooted Gromov-Hausdorff closeness to an interval, a sharp uniform estimate for the rearranged torsion profile, and explicit uniform estimates for the torsion function on the original graph. All estimates allow arbitrary edge counts and arbitrarily short positive edges.
Author: Xu, Zihuan
Institution: Capital Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225204730