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More on the full Brouwer Laplacian spectrum conjecture

Xiaodan Chen, Junwei Zi

TL;DR

This work advances the understanding of the full Brouwer Laplacian spectrum conjecture by (i) presenting a concise version and establishing it for two broad graph families: spanning subgraphs of complete split graphs and $c$-cyclic graphs with $c\in\{0,1,2\}$, with precise equality characterizations linked to threshold graphs; (ii) developing Nordhaus–Gaddum-type bounds for the sums of the top Laplacian eigenvalues and proving several partial results under density and degree constraints; and (iii) outlining connections to threshold and split graph structures via majorization and Ferrers diagrams. The results strengthen the conjecture's validity in structurally constrained graph classes and lay groundwork for broader generalization, while highlighting remaining hard cases and potential higher-dimensional extensions.

Abstract

Brouwer conjectured that the sum of the first $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for each $k\in \{1,2,\cdots,n\}$, which has come to be known as Brouwer's conjecture. Recently, Li and Guo further considered the case when the equalities hold in these conjectured inequalities, and proposed the full version of Brouwer's conjecture. In this paper, we first present a concise version of the full Brouwer's conjecture. Then we show that the full Brouwer's conjecture holds for two families of spanning subgraphs of complete split graphs and for $c$-cyclic graphs with $c\in\{0,1,2\}$. We also consider the Nordhaus-Gaddum version of the full Brouwer's conjecture and present partial solutions to it.

More on the full Brouwer Laplacian spectrum conjecture

TL;DR

This work advances the understanding of the full Brouwer Laplacian spectrum conjecture by (i) presenting a concise version and establishing it for two broad graph families: spanning subgraphs of complete split graphs and -cyclic graphs with , with precise equality characterizations linked to threshold graphs; (ii) developing Nordhaus–Gaddum-type bounds for the sums of the top Laplacian eigenvalues and proving several partial results under density and degree constraints; and (iii) outlining connections to threshold and split graph structures via majorization and Ferrers diagrams. The results strengthen the conjecture's validity in structurally constrained graph classes and lay groundwork for broader generalization, while highlighting remaining hard cases and potential higher-dimensional extensions.

Abstract

Brouwer conjectured that the sum of the first largest Laplacian eigenvalues of an -vertex graph is less than or equal to the number of its edges plus for each , which has come to be known as Brouwer's conjecture. Recently, Li and Guo further considered the case when the equalities hold in these conjectured inequalities, and proposed the full version of Brouwer's conjecture. In this paper, we first present a concise version of the full Brouwer's conjecture. Then we show that the full Brouwer's conjecture holds for two families of spanning subgraphs of complete split graphs and for -cyclic graphs with . We also consider the Nordhaus-Gaddum version of the full Brouwer's conjecture and present partial solutions to it.

Paper Structure

This paper contains 7 sections, 20 theorems, 81 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Spanning subgraphs of complete split graphs
  5. $c$-cyclic graphs with $c\in\{0,1,2\}$
  6. Nordhaus-Gaddum-type results for $s_k$
  7. Concluding remark

Key Result

Theorem 1.1

For any graph $G$ on $n$ vertices with (non-increasing) Laplacian eigenvalue sequence $\mu(G):=(\mu_1,\mu_2,\cdots,\mu_n)$ and conjugated degree sequence $d^*(G):=(d_1^*,d_2^*,\cdots,d_n^*)$, $\mu(G)$ is majorized by $d^*(G)$, namely, Moreover, the equality holds if and only if $G$ is a threshold graph.

Figures (9)

  • Figure 1: The Ferrers-Sylvester diagram of a threshold graph.
  • Figure 2: The $\infty$-graph $\infty(p,l,q)$ and $\theta$-graph $\theta(p,l,q)$.
  • Figure 3: The tree $T_n^{i}$, $0\leq i\leq (n-1)/2$.
  • Figure 4: The graph $G(s,n-2-s), 0\leq s\leq n-3$.
  • Figure 5: Parts $X$, $Y$ and $Z$ in the Ferrers-Sylvester diagram of a threshold graph $G$.
  • ...and 4 more figures

Theorems & Definitions (33)

  • Theorem 1.1: Grone-Merris-Bai theorem
  • Conjecture 1.2: Brouwer's conjecture
  • Conjecture 1.3: The full Brouwer's conjecture
  • Conjecture 1.4: A concise version of the full Brouwer's conjecture
  • Conjecture 1.5
  • Lemma 2.1: see Merris
  • Lemma 2.2: see Du
  • Lemma 2.3: Fan's inequality Fan
  • Lemma 2.4: see Wang
  • Lemma 2.5: see Zhou
  • ...and 23 more