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

Source abstract

Let $G$ be a simple graph of order $n\ge2$. Let $e(G)$ and $ω(G)$ denote its number of edges and clique number, respectively. Let $λ_1(G)\ge\cdots\geλ_n(G)$ be its adjacency eigenvalues. In 2007, Bollobás and Nikiforov conjectured that every graph $G\neq K_n$ satisfies $λ_1^2(G)+λ_2^2(G)\le2\left(1-\frac1{ω(G)}\right)e(G).$ This conjecture was recently completely resolved by Coutinho, Liu, Spier, Tang and Zhang. For every edge $e\in E(G)$, let $c_G(e)$ denote the maximum order of a clique containing $e$. Kannan, Kumar and Pragada proposed the edge-localized conjecture $λ_1^2(G)+λ_2^2(G)\le2\sum_{e\in E(G)}\left(1-\frac1{c_G(e)}\right)$ for every graph $G\neq K_n$. 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.