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In this paper, we obtain monotonicity of Steklov eigenvalues on graphs whichas a special case on trees extends the results of He-Hua [Calc. Var. PartialDifferential Equations 61 (2022), no. 3, Paper No. 101, arXiv: 2103.07696] tohigher Steklov eigenvalues and gives affirmative answers to two problemsproposed in He-Hua [arXiv: 2103.07696]. As applications of the monotonicity ofSteklov eigenvalues, we obtain some estimates for Steklov eigenvalues on treesgeneralizing the isodiametric estimate for the first positive Stekloveigenvalues on trees in He-Hua [arXiv:2011.11014].

Monotonicity of Steklov eigenvalues on graphs and applications