Maximum spectral sum of graphs

TL;DR

This paper proves a conjecture that the sum of the two largest adjacency eigenvalues of any graph is at most 8/7 times the number of vertices, using advanced mathematical tools.

Contribution

It confirms the conjecture on the maximum spectral sum of graphs, introducing novel methods from multiple mathematical disciplines.

Findings

The spectral sum rac{8}{7}n for any graph G.

New bounds established for adjacency eigenvalues.

Techniques applicable to other spectral graph problems.

Abstract

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum \[ \lambda_1(G) + \lambda_2(G)\le \frac{8}{7}n \] 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.