Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths
Abstract
We study torsional rigidity on finite compact connected metric graphs with real δ-vertex strengths in the regime of strictly positive torsion. For a fixed total length L>0, let I+ be the sum of the positive strengths and I- the sum of the magnitudes of the negative strengths, with I+>I-. We determine the sharp infimum and the exact threshold for a finite universal supremum. The supremum is finite precisely when I-<I+/(1+LI+). In this regime, an interval with strengths I+ and −I- at opposite endpoints attains the sharp Saint--Venant bound. At and beyond the threshold, torsional rigidity is unbounded even among three-vertex paths with the same data and strictly positive torsion. The sharp infimum is L2/(I+−I-) throughout the range I+>I-≥0. It is not attained by any finite graph and is approached by graphs with two vertices joined by increasingly many parallel edges of equal length, preserving both strength totals separately. A vertex-matrix formulation and explicit path examples clarify the relation between strength placement, spectral positivity, and solvability.
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BibTeX
@misc{ozcan2026sharp,
title = {{Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths}},
author = {Sedef Özcan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38330},
eprint = {2610.38330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38330},
}
RIS
TY - UNPB
TI - Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths
AU - Sedef Özcan
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38330
UR - https://arxiv.org/abs/2610.38330
AB - We study torsional rigidity on finite compact connected metric graphs with real δ-vertex strengths in the regime of strictly positive torsion. For a fixed total length L>0, let I_+ be the sum of the positive strengths and I_- the sum of the magnitudes of the negative strengths, with I_+>I_-. We determine the sharp infimum and the exact threshold for a finite universal supremum. The supremum is finite precisely when I_-<I_+/(1+LI_+). In this regime, an interval with strengths I_+ and −I_- at opposite endpoints attains the sharp Saint--Venant bound. At and beyond the threshold, torsional rigidity is unbounded even among three-vertex paths with the same data and strictly positive torsion. The sharp infimum is L^2/(I_+−I_-) throughout the range I_+>I_-≥0. It is not attained by any finite graph and is approached by graphs with two vertices joined by increasingly many parallel edges of equal length, preserving both strength totals separately. A vertex-matrix formulation and explicit path examples clarify the relation between strength placement, spectral positivity, and solvability.
ER -