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

Nikiforov (2002) 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 in 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.