Spectral Extremal Results on Trees

Let spex(n,F) be the maximum spectral radius over all F-free graphs of order n, and SPEX(n,F) be the family of F-free graphs of order n with spectral radius equal to spex(n,F). Given integers $n,k,p$ with $n>k>0$ and 0≤ p≤ (n-k)/2, let Sn,kᵖ be the graph obtained from Kₖ∇(n-k)K₁ by embedding p independent edges within its independent set, where '∇' means the join product. For n≥≥ 4, let Gn,=Sn,(-2)/2⁰ if is even, and Gn,=Sn,(-3)/2¹ if is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for ≥ 6 and sufficiently large n, if ρ(G)≥ ρ(Gn,), then G contains all trees of order unless G=Gn,. They further posed a problem to study spex(n,F) for various specific trees F. Fix a tree F of order ≥ 6, let A and B be two partite sets of F with |A|≤ |B|, and set $q=|A|-1$. We first show that any graph in SPEX(n,F) contains a spanning subgraph Kq,n-q for q≥ 1 and sufficiently large n. Consequently, ρ(Kq,n-q)≤ spex(n,F)≤ ρ(Gn,), we further respectively characterize all trees F with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of Gn,.