Universal clique-spectral extremal conjecture for even-order trees

Determine, for every fixed k≥2 and r≥2 and all sufficiently large n, whether the universal r-clique spectral extremal quantity for graphs on n vertices that omit at least one tree of order 2k+2 satisfies Λ_r(n,2k+2)=ρ_r(S_{n,k}) for 2≤r≤k+1 and Λ_r(n,2k+2)=ρ_r(K_{2k+1})=\binom{2k}{r−1} for k+2≤r≤2k+1; additionally, establish uniqueness of S_{n,k} in the first range and show that every extremal graph in the second range contains K_{2k+1} as a component.

Background

The quantity Λr(n,t) maximizes the r-clique spectral radius among n-vertex graphs that fail to contain at least one t-vertex tree. The proposed even-order conjecture interpolates between split graphs S{n,k}, which have growing clique-spectral radius for small r, and complete components K_{2k+1}, which dominate for larger r. The paper proves only the large-clique component regime under more restrictive parameter conditions.