The Bollob\'{a}s--Nikiforov Conjecture for Complete Multipartite Graphs and Dense $K_4$-Free Graphs

TL;DR

This paper proves the Bollobás–Nikiforov conjecture for complete multipartite graphs and dense $K_4$-free graphs, using spectral methods and stability analysis, and identifies obstacles for certain approaches.

Contribution

It establishes the conjecture for all complete multipartite graphs with more vertices than parts and for dense $K_4$-free graphs, providing spectral and structural insights.

Findings

Proves the conjecture for all complete multipartite graphs with n > r.

Shows that for such graphs, the second eigenvalue is zero when vertices exceed parts.

Demonstrates stability of $K_4$-free graphs near the Turán maximum spectral radius.

Abstract

The Bollob\'as--Nikiforov conjecture asserts that for any graph $G \neq K_n$ with $m$ edges and clique number $\omega(G)$, \[ \lambda_1^2(G) + \lambda_2^2(G) \;\leq\; 2\!\left(1 - \frac{1}{\omega(G)}\right)m, \] where $\lambda_1(G) \geq \lambda_2(G) \geq \cdots \geq \lambda_n(G)$ are the adjacency eigenvalues of $G$. We prove the conjecture for all complete multipartite graphs $K_{n_1,\ldots,n_r}$ with $n_1 + \cdots + n_r > r$. The proof computes the full spectrum via a secular equation, establishes that $\lambda_2 = 0$ whenever the graph has more vertices than parts, and then applies Nikiforov's spectral Tur\'an theorem; equality holds if and only if all parts have equal size. We also prove a stability result for $K_4$-free graphs whose spectral radius is near the Tur\'an maximum: such graphs are structurally close to the balanced complete tripartite graph, and as a consequence the conjecture holds for all $K_4$-free graphs with $m = \Omega(n^2)$ when $n$ is sufficiently large. Finally, we identify the precise obstruction preventing a Hoffman-bound approach from settling the conjecture for $K_4$-free graphs with independence number $\alpha(G) \geq n/3$.