Proves m_L(T)(1) ≤ (n-5)/6 for reduced trees on n≥7 vertices meeting stated conditions and m_L(G)(1) ≤ c + (n-2)/4 for connected reduced graphs on n≥6, with extremal cases fully characterized.
Du, The multiplicity of eigenvalues of Aα of trees, Ann
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Multiplicity of Laplacian eigenvalue 1 of a graph
Proves m_L(T)(1) ≤ (n-5)/6 for reduced trees on n≥7 vertices meeting stated conditions and m_L(G)(1) ≤ c + (n-2)/4 for connected reduced graphs on n≥6, with extremal cases fully characterized.