Abstract: Let EX(n,H) and SPEX(n,H) denote the families of H-free graphs of order n with maximum size and maximum spectral radius, respectively. A graph H is called spectralconsistent if SPEX(n,H)⊆EX(n,H) holds for all sufficiently large n. Cioaba, Desai, and Tait put forward the following conjecture: Let H be a graph for which every graph in EX(n,H) is a Turán graph plus O(1) edges. Then H is spectral-consistent. Wang, Kang, and Xue settled this conjecture. Recently, Liu and Ning raised a general problem in spectral extremal graph theory: Characterize all graphs that are spectral-consistent. In this paper, we establish that for any finite graph H, if its decomposition family is matching-good, then H is necessarily spectral-consistent. Notably, this structural condition is strictly weaker than the condition for spectral-consistency established by Wang, Kang, and Xue in their earlier work, thereby broadening the class of graphs known to satisfy the spectral-consistency property. Our main result enables us to fully characterize spectral consistency for several important families of forbidden graphs H, including generalized color-critical graphs, odd-balloonings of nontrivial trees and complete bipartite graphs Ks,t with t≥s≥2 and (s,t) ̸=(2,3), as well as edge blow-ups of non-bipartite graphs and even cycles. Furthermore, we present a streamlined proof of the Chen–Lei--Li theorem on the spectral-consistency of disjoint unions of suspensions of unions of color-critical graphs. Finally, we propose an open problem to motivate future research in this area.

Cite: Fang L. , Goryainov S. , Krotov D. , Lin H. , Zhai М.
The spectral Turán problem: Characterizing spectral-consistent graphs
Discrete Mathematics. 2027. V.350. N3. 115496 :1-15. DOI: 10.1016/j.disc.2026.115496

Dates:

Submitted:	Mar 21, 2026
Accepted:	Oct 20, 2026

Identifiers: No identifiers

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