The spectral Turán problem: Characterizing spectral-consistent graphs

Abstract

Let $\mathrm{EX}(n,H)$ and $\mathrm{SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if $\mathrm{SPEX}(n,H)\subseteq \mathrm{EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioabă, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be any graph such that the graphs in $\mathrm{EX}(n,H)$ are Turán graph plus $O(1)$ edges. Then $H$ is spectral-consistent. Wang, Kang and Xue [J. Combin. Theory Ser. B 159 (2023) 20--41] confirmed this conjecture, along with a stronger result. 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 the spectral-consistency for several important families of forbidden graphs \(H\), including generalized color-critical graphs, odd-ballooning of trees and complete bipartite graphs, as well as edge blow-up of non-bipartite graphs and certain special bipartite graphs. Furthermore, we present a streamlined proof for an existing spectral-consistency result due to Chen, Lei, and Li, simplifying their original argument. Finally, we propose several open problems to motivate future research in this area.

The second version has been updated to incorporate new results, as well as a Concluding Remarks section presenting two open problems