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| Paper IPM / M / 11550 |
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Let G be a graph of order n with signless Laplacian eigenvalues q1,...,qn and Laplacian eigenvalues μ1,...,μn. It is proved that for any real number α with 0 < α\leqslant1 or 2\leqslant α < 3, the inequality q1α+ ... + qnα\geqslant μ1α +... + μnαholds, and for any real number β with 1 < β < 2, the inequality q1β+ ...+qnβ \leqslant μ1β +...+μnβ holds. In both inequalities, the equality is attained (for α ∉ {1,2}) if and only if G is biparite.
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