Extremal chromatic bounds for distance Laplacian eigenvalues

Abstract: For a connected simple graph $G$ on $n$ vertices with chromatic number $χ$, the distance Laplacian matrix is $\DL( G)=\mathrm{diag}(\mathrm{Tr}{ G}(v_1),\dots,\mathrm{Tr}{ G}(v_n)) - D( G)$, where $D( G)$ is the distance matrix and $\mathrm{Tr}{ G}(v)=\sum{u\in V( G)} d_{ G}(u,v)$ 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 $\partial^{L}_1( G)\ge n+\ceil{\frac{n}χ}$ 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 the distribution theorems of Aouchiche--Hansen (Filomat, 2017) and Pirzada--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 all minimizers. Several numerical examples are included along with pictorial representations.

• The paper establishes a color-class majorization theorem that links optimal color-class sizes with lower bounds on the largest distance Laplacian eigenvalues.
• The study connects structural twin mechanisms in cliques and independent sets to eigenvalue multiplicities above a computed chromatic spectral threshold.
• The paper fully characterizes extremal graphs with fixed chromatic number, shedding light on spectral bounds relevant to network analysis and algorithmic coloring.

Extremal Chromatic Bounds for Distance Laplacian Eigenvalues

Introduction and Context

This paper investigates the interplay between the chromatic number $\chi(G)$ of a connected simple graph $G$ and the spectrum of its distance Laplacian matrix $DL(G)$. Extending and sharpening recent work on chromatic bounds for the distance Laplacian spectral radius, the author establishes majorization and multiplicity principles governing the distribution of large eigenvalues in terms of optimal color-class sizes. The results both unify and strengthen previous distribution theorems, and provide extremal characterizations of minimizers at fixed chromatic number.

Distance-based matrix analysis extends classical spectral graph theory to global combinatorial invariants, producing tools that are sensitive to network structure beyond simple adjacency. The largest distance Laplacian eigenvalue, the spectral radius $\partial^L_1(G)$, in particular, encodes information about both the distance distribution and chromatic constraints, and its study is motivated by applications ranging from network heterogeneity to algorithmic coloring.

Main Theoretical Contributions

Color-Class Majorization Principle

The core contribution of the paper is a color-class majorization theorem: For a graph $G$ with chromatic number $\chi$, consider an optimal coloring with color-class sizes $\ell_1 \geq \ell_2 \geq \cdots \geq \ell_\chi$. Then, denoting $b_\chi = n + n/\chi$ as the chromatic spectral threshold, the author proves that at least $\ell_1-1$ of the largest distance Laplacian eigenvalues of $G$ satisfy $G$0, for $G$1. This dominant color-class effect extends blockwise: for the $G$2th largest color class of size $G$3, the next $G$4 eigenvalues are at least $G$5.

This result refines earlier distribution theorems by Aouchiche--Hansen and Pirzada--Khan by shifting the focus from simply counting eigenvalues above $G$6 to an explicit lower bound based on color-class sizes. The majorization result is sharp for complete multipartite graphs, where color classes correspond to partition parts.

Multiplicity Mechanisms: Clique and Independent Set Twins

A further structural result connects distance Laplacian eigenvalue multiplicities to the presence of twin classes (cliques or independent sets with identical neighborhood structure). Specifically:

• If $G$7 contains a clique $G$8 of size $G$9 with identical external neighborhoods, then $DL(G)$0 has an eigenvalue at least $DL(G)$1 (with $DL(G)$2 the shared neighborhood), with multiplicity at least $DL(G)$3, and if $DL(G)$4, this eigenvalue exceeds $DL(G)$5.
• Analogously, an independent set of size $DL(G)$6 with common neighborhood $DL(G)$7 forces an eigenvalue at least $DL(G)$8 of multiplicity $DL(G)$9 above $\partial^L_1(G)$0 when $\partial^L_1(G)$1.

These criteria improve on previous results by replacing minimal class size conditions with explicit neighborhood-compression parameters, demonstrating that even small twin structures can force several large eigenvalues when their external connection is sparse.

Extremal Graphs for Fixed Chromatic Number

The extremal problem for minimizing the spectral radius $\partial^L_1(G)$2 among all connected graphs $\partial^L_1(G)$3 with fixed chromatic number $\partial^L_1(G)$4 is fully resolved: the minimum is attained exactly for complete $\partial^L_1(G)$5-partite graphs with balanced part sizes, and the minimum value is $\partial^L_1(G)$6. The proof employs the edge-deletion monotonicity of the distance Laplacian, reducing the minimization to edge-maximal colorings, and utilizes the explicit spectrum of multipartite graphs.

For $\partial^L_1(G)$7, this characterization is unique up to isomorphism, with the part-size vector determined precisely.

Unified Distribution Bounds with Diameter Sensitivity

The paper combines the color-class majorization principle with classical diameter distribution theorems to yield improved lower bounds on the number of eigenvalues above $\partial^L_1(G)$8. For instance, if $\partial^L_1(G)$9, then at least $G$0 eigenvalues satisfy $G$1, subsuming earlier results for large $G$2 or large diameter.

Numerical Results and Example Graphs

Extensive numerical examples validate the theoretical bounds, applying them to complete multipartite graphs, trees, paths, cycles, and graphs constructed to illustrate twin clique and independent set phenomena. The results confirm that the majorization bound is tight for certain extremal graphs, and that diameter-based lower bounds are indeed attained. For example, the double star $G$3 and graphs constructed with twin independent sets show how structural features enforce eigenvalue multiplicities and placements predicted by the theorems.

Theoretical and Practical Implications

The study demonstrates that the distribution of large distance Laplacian eigenvalues is tightly controlled by coloring structure and the sizes of dominant color classes, extending the role of the chromatic number from a global combinatorial invariant to a spectral certificate. This has implications for network analysis, spectral partitioning, and large-scale algorithmic coloring, particularly in bounding network heterogeneity and certifying complexity measures from easily computable invariants.

From a theoretical standpoint, the work unifies chromatic, diameter, and structural twin class effects in spectral distribution theorems, and paves the way for analogous results in variants such as the normalized distance Laplacian or signless Laplacian.

Future Directions

Possible extensions include:

• Sharpening the diameter-sensitive bounds for small and large $G$4;
• Characterizing the equality cases in the majorization principle beyond complete multipartite graphs;
• Adapting the majorization technique to other distance-based or normalized Laplacian matrices and other spectral invariants;
• Investigating applications in addressing large-scale heterogeneity in practical networks using spectral bounds derived from coloring structure.

Conclusion

This paper makes significant advances in the spectral analysis of the distance Laplacian relative to chromatic constraints, producing precise majorization theorems, multiplicity criteria for structural twins, and extremal characterizations at fixed chromatic number. The developed principles sharpen classical bounds by leveraging explicit color-class information, and suggest fruitful directions for both combinatorial spectral theory and distance-based network analysis.