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arXiv · 2610.38330

Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths

Sedef Özcan
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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 \(I_- I_-\ge0\). 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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Sedef Özcan. 2026-10-28. Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths. https://arxiv.org/abs/2610.38330

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