Strong spectral stabilities for C₂ₖ₊₁-free graphs

A stability result due to Ren, Wang, Wang and Yang [SIAM J. Discrete Math. 38 (2024)] shows that if 3≤ r ≤ 2k and n≥ 318 (r-2)²k, and G is a C₂ₖ₊₁-free graph on n vertices with e(G)≥ (n-r+1)²/4 +r 2, then G can be made bipartite by deleting at most $r-2$ vertices. Using a different method, we give a linear bound on n in terms of k and show a stronger structural result, which roughly says that G can be obtained from a large bipartite graph by suspending some small graphs that the total number of vertices is at most $r-2$. This improves a result of Yan and Peng (2024) by weakening the requirement on n and k. As a direct corollary, we obtain a tight upper bound on the size of an n-vertex C₂ₖ₊₁-free graph with chromatic number χ(G)≥ r for every r≤ 2k. The second part of this paper concerns the spectral extremal problem for C₂ₖ₊₁-free graphs. We denote by $λ(G)$ the spectral radius of the adjacency matrix of a graph G. Let Tn-r+1,2∘ Kᵣ be the graph obtained by identifying a vertex of the complete graph Kᵣ and a vertex of the smaller partite set of the bipartite Turán graph Tn-r+1 ,2. Using the spectral techniques, we prove that if 3≤ r≤ 2k and n≥ 712k, and G is an n-vertex C₂ₖ₊₁-free graph with chromatic number χ(G) ≥ r, then λ(G)≤ λ(Tn-r+1,2∘ Kᵣ), where the equality holds if and only if G=Tn-r+1,2∘ Kᵣ. Our result not only extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a result of Zhang and Zhao [Discrete Math. 346 (2023)], but also provides the first solution to the spectral extremal problem for F-free graphs with high chromatic number.