A Brualdi–Hoffman–Turán problem on cycles

Brualdi–Hoffman–Turán-type problem asks what is the maximum spectral radius λ1(G) of an H-free graph G on m edges? This problem gives a spectral perspective on the existence of a subgraph H. A significant result, due to Nikiforov, states that λ1(G)≤2m(1−1r) for every Kr+1-free graph G (Nikiforov, 2002). Bollobás and Nikiforov further conjectured λ12(G)+λ22(G)≤2m(1−1r) for every Kr+1-free graph G (Bollobás and Nikiforov, 2007). Let Ck+ denote the graph obtained from a k-cycle by adding a chord between two vertices of distance two. Zhai, Lin and Shu conjectured that for k≥2 and m sufficiently large, if G is a C2k+1-free or C2k+2-free graph, then λ1(G)≤k−1+4m−k2+12, with equality if and only if G≅Kk∇(mk−k−12)K1 (Zhai et al., 2021). This conjecture was also included in a survey of Liu and Ning as one of twenty unsolved problems in spectral graph theory. Recently, Y.T. Li posed a stronger conjecture, which states that the above spectral bound holds for C2k+1+-free and C2k+2+-free graphs. In this paper, we confirm these two conjectures by using k-core method and spectral techniques. This presents a new approach to study Brualdi–Hoffman–Turán problems