A characterization of equality in the edge-localized Bollobás-Nikiforov inequality

Let G be a simple graph of order n≥2. Let $e(G)$ and $ω(G)$ denote its number of edges and clique number, respectively. Let λ₁(G)≥⋯≥λₙ(G) be its adjacency eigenvalues. In 2007, Bollobás and Nikiforov conjectured that every graph G≠ Kₙ satisfies λ₁²(G)+λ₂²(G)≤2(1-1ω(G))e(G). This conjecture was recently completely resolved by Coutinho, Liu, Spier, Tang and Zhang. For every edge e∈ E(G), let cG(e) denote the maximum order of a clique containing e. Kannan, Kumar and Pragada proposed the edge-localized conjecture λ₁²(G)+λ₂²(G)≤2∑e∈ E(G)(1-1cG(e)) for every graph G≠ Kₙ. This conjecture was subsequently proved by Shaik. In this paper, we provide complete structural characterizations of the equality cases in both the Bollobás-Nikiforov inequality and its edge-localized version.