On spectral geometry of metric graphs: Diameter bounds, cycles, and rigidity
Keywords:
Spectral gap, Diameter bounds, Cycles, Spectral rigidity, Phase transitionsAbstract
We study how the spectral gap λ1(G) of a compact metric graph equipped with the Kirchhoff Laplacian depends on its geometric and topological features. First, we examine λ1(G) ≥ π2/ D2 and demonstrate that equality is not unique to interval graphs: a single cycle graph C of length l (with diameter D = l/2) satisfies λ1(C) = 4π2/ l2 = π2/ D2, providing a non-interval equality case. Furthermore, we analyze basic topological operations, proving that edge subdivision preserves the spectrum, pendant edge contraction strictly increases λ1(G) unless a first eigenfunction vanishes at the attachment vertex (in which case λ1(G) is invariant), and function addition yields a weighted average of Rayleigh quotients rather than an additive sum even under mutual H1-orthogonality.
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Copyright (c) 2027 Kajal Rani, Junaid Nisar, Haribhau R. Bhapkar, Shabir Ahmad Mir, Gurpreet Kaur (Author)
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