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 rho(G) <= root 2m(1 - 1/r), which implies that G contains C-3 if rho(G) > root m. In this paper, we follow this direction in determining which subgraphs will be contained in G if rho(G) > f(m), where f(m) similar to root m as m -> infinity. We first show that if rho(G) >= root m, then G contains K-2,K-r+1 unless G is a star; and G contains either C-3(+) or C-4(+) unless G is a complete bipartite graph, where C-t(+) denotes the graph obtained from C-t and C-3 by identifying an edge. Secondly, we prove that if rho(G) >= m - 4, then G contains pentagon and hexagon unless G is a book; and if p(G) >= 1/2 + root m - 3/4, then G contains pentagon and hexagon unless G is a book; and if rho(G) > 1/2(k - 1/2) + root m + 1/4(k - 1/2)(2), then G contains C-t for every t <= 2k +2. In the end, some related conjectures are provided for further research. (C) 2021 Elsevier Ltd. All rights reserved.