LAUSR

Abstract Let A ( G ) be the adjacency matrix of a graph G and ρ ( G ) be its spectral radius. Given a graph H and a family F of graphs, let e x s p ( n , H ; F ) = max ⁡ { ρ ( G ) | | V ( G ) | = n , H ⊆ G , G does not contain any graph of F } . Let S 2 k − 1 ( K s , t ) be the graph obtained by replacing an edge of K s , t with a copy of P 2 k + 1 , where k ≥ 2 . In this paper, we show that e x s p ( n , C 2 k + 3 ; { C 3 , C 5 , … , C 2 k + 1 } ) = ρ ( S 2 k − 1 ( K ⌈ n − 2 k + 1 2 ⌉ , ⌊ n − 2 k + 1 2 ⌋ ) ) and the unique extremal graph is S 2 k − 1 ( K ⌈ n − 2 k + 1 2 ⌉ , ⌊ n − 2 k + 1 2 ⌋ ) , which solves a question proposed in [Eigenvalues and triangles in graphs, Comb. Probab. Comput. 30 (2021) 258–270].