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Maximum spectral sum of graphs

Hitesh Kumar, Lele Liu, Hermie Monterde, Shivaramakrishna Pragada, Michael Tait

Abstract

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $λ_1(G) + λ_2(G)$, where $λ_1(G)$ (resp. $λ_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum \[ λ_1(G) + λ_2(G)\le \frac{8}{7}n \] for any graph $G$. We prove this conjecture by combining tools from the theory of graph limits, convex geometry, exterior algebra and convex optimization. The techniques developed are of independent interest.

Maximum spectral sum of graphs

Abstract

For a graph of order , the spectral sum of is defined to be the sum , where (resp. ) is the largest (resp. second largest) adjacency eigenvalue of . Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum for any graph . We prove this conjecture by combining tools from the theory of graph limits, convex geometry, exterior algebra and convex optimization. The techniques developed are of independent interest.

Paper Structure

This paper contains 13 sections, 23 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotic result suffices
  3. Enter graphons
  4. Proof of Theorem \ref{['thm:upper_bound_8_over_7_graphon']}
  5. Adjacency criterion
  6. Ellipse equation for $f$ and $g$
  7. $f$ and $g$ are step functions with at most $6$ steps
  8. Monotonicity of $f$ and $g$
  9. Back to (weighted) graphs
  10. Structure of $G^*$
  11. Exterior algebra
  12. Matrix sum of squares
  13. Concluding remarks

Key Result

Theorem 1.1

For any graph $G\in \mathcal{G}(n)$, we have $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Possibilities for $G^*$. The vertex labeled by $i$ in the figure corresponds to $U_i$.

Theorems & Definitions (47)

  • Theorem 1.1: Ebrahimi_Mohar_Nikiforov_Ahmady_2008
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 3.1: Min-Max Theorem
  • Lemma 3.2
  • Theorem 3.3: cf. Borgs_Chayes_Lovasz_Sos_Vesztergombi_2012 or Ore_1962
  • Lemma 3.4
  • Theorem 3.5
  • ...and 37 more