The maximum spectral radius of C4-free graphs of given order and size
Suppose that G is a graph with n vertices and m edges, and let μ be the spectral radius of its adjacency matrix. Recently we showed that if G has no 4-cycle, then μ2 - μ ≤ n - 1, with equality if and only if G is the friendship graph. Here we prove that if m ≥ 9 and G has no 4-cycle, then μ2 ≤ m, with equality if G is a star. For 4 ≤ m ≤ 8 this assertion fails. © 2009 Elsevier Inc. All rights reserved.
Linear Algebra and Its Applications
Nikiforov, V. (2009). The maximum spectral radius of C4-free graphs of given order and size. Linear Algebra and Its Applications, 430 (2022-11-12), 2898-2905. https://doi.org/10.1016/j.laa.2009.01.002