We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.

翻译：我们证明了对于所有k∈{1,...,n}，图或箭图的基尔霍夫拉普拉斯算子K的每个特征值l(k)均满足上界l(k) ≤ d(k)+d(k-1)。此处l(1),...,l(n)是K特征值的非递减序列，d(1),..,d(n)是顶点度数的非递减序列，并附加假设d(0)=0。我们还证明了在一般情况下，弱Brouwer-Haemers下界d(k) + (n-k)对箭图基尔霍夫矩阵的所有特征值l(k)均成立。