For a connected simple graph $G$ on $n$ vertices with chromatic number $χ$, the distance Laplacian matrix is $\DL(G)=\operatorname{diag}(\Tr_G(v_1),\dots,\Tr_G(v_n))-D(G)$, where $D(G)$ is the distance matrix and $\Tr_G(v)=\sum_{u\in V(G)} d_G(u,v)$ is the transmission. The eigenvalues of $\DL(G)$ are ordered as $\partial^{L}_1(G)\ge \partial^{L}_2(G)\ge \cdots \ge \partial^{L}_n(G)=0$. Building on the chromatic lower bound $\partial^{L}_1(G)\ge n+\ceil{n/χ}$ and subsequent developments, we prove a \emph{color-class majorization principle}: if $(\ell_1,\dots,\ell_χ)$ are the color-class sizes in an optimal $χ$-coloring with $\ell_1\ge\cdots\ge\ell_χ$, then the first $\ell_1-1$ distance Laplacian eigenvalues satisfy $\partial^{L}_i(G)\ge n+\ell_1$, for $1\le i\le \ell_1-1$. This gives sharp lower bounds on the number of eigenvalues above the chromatic threshold $b_χ=n+\ceil{n/χ}$, thereby refining distribution theorems of [Aouchiche and Hansen, Filomat, 2017] and [Pirzada and Khan LAA, 2021]. We further refine clique/independent-set based multiplicity results by deriving explicit chromatic criteria in terms of neighborhood compression, and we generalize the extremal problem for minimum $\partial^{L}_1$ at fixed chromatic number by characterizing the balanced complete multipartite minimizers. Finally, we present a Ky Fan type result, and complement-component consequences of the majorization principle.

翻译：对于顶点数为$n$、色数为$\chi$的连通简单图$G$，其距离拉普拉斯矩阵定义为$\DL(G)=\operatorname{diag}(\Tr_G(v_1),\dots,\Tr_G(v_n))-D(G)$，其中$D(G)$是距离矩阵，$\Tr_G(v)=\sum_{u\in V(G)} d_G(u,v)$为传递度。$\DL(G)$的特征值按降序排列为$\partial^{L}_1(G)\ge \partial^{L}_2(G)\ge \cdots \ge \partial^{L}_n(G)=0$。基于色数下界$\partial^{L}_1(G)\ge n+\ceil{n/\chi}$及后续发展，我们证明了\textbf{色类优控原理}：若$(\ell_1,\dots,\ell_\chi)$为最优$\chi$着色中的色类大小且满足$\ell_1\ge\cdots\ge\ell_\chi$，则前$\ell_1-1$个距离拉普拉斯特征值满足$\partial^{L}_i(G)\ge n+\ell_1$（$1\le i\le \ell_1-1$）。该结果给出了色数阈值$b_\chi=n+\ceil{n/\chi}$以上特征值个数的尖锐下界，从而改进了[Aouchiche and Hansen, Filomat, 2017]与[Pirzada and Khan LAA, 2021]的分布定理。我们进一步通过邻域压缩导出显式色数判别准则，完善了基于团/独立集的重数结果，并刻画了最小$\partial^{L}_1$在固定色数下的极值问题——平衡完全多部图是唯一极小元。最后，我们提出Ky Fan型结果以及优控原理的补分量推论。