On the multiplicity of positive eigenvalues of a graph

Let T be a tree on n(≥7) vertices with λ as a positive eigenvalue of multiplicity k. If λ2≥2 is an integer, then we prove that k≤⌊n−43⌋ and all extremal graphs attaining the upper bound are characterized. This result revises and improves the main conclusion of Wong, Zhou and Tian (2020). Moreover, applying this result we investigate the eigenvalue multiplicity of unicyclic graphs. Let G be a unicyclic graph of order n(≥11), which contains λ (λ2≥2 is an integer) as a positive eigenvalue of multiplicity m. Then it is proved that m≤⌊n−23⌋, and all extremal graphs attaining the upper bound are determined. These two upper bounds improve the conclusions of Rowlinson (2010, 2011), respectively.