Extremal chromatic bounds for distance Laplacian eigenvalues
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| Format: | Preprint |
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2026
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| _version_ | 1866917496713379840 |
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| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal chromatic bounds for distance Laplacian eigenvalues Rather, Bilal Ahmad Combinatorics Discrete Mathematics 05C50, 05C12, 15A18 F.2.2 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. |
| title | Extremal chromatic bounds for distance Laplacian eigenvalues |
| topic | Combinatorics Discrete Mathematics 05C50, 05C12, 15A18 F.2.2 |
| url | https://arxiv.org/abs/2604.10785 |