Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs

This is the authors' abstract. We don't add key points for this paper.

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if G is Kᵣ₊₁-free then the spectral radius ρ(G)≤√2m(1-1/r), which implies that G contains C₃ if ρ(G)>√m. In this paper, we follow this direction on determining which subgraphs will be contained in G if $ρ(G)> f(m)$, where f(m)~√m as m→ ∞. We first show that if ρ(G)≥ √m, then G contains K2,r+1 unless G is a star; and G contains either C₃⁺ or C₄⁺ unless G is a complete bipartite graph, where Cₜ⁺ denotes the graph obtained from Cₜ and C₃ by identifying an edge. Secondly, we prove that if ρ(G)≥12+√m-34, then G contains pentagon and hexagon unless G is a book; and if ρ(G)>12(k-12)+√m+14(k-12)², then G contains Cₜ for every t≤ 2k+2. In the end, some related conjectures are provided for further research.