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

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 $\alpha(G) \geq n/3$.