The $α$-spectral Turán type problems for graphs
Jiadong Wu, Yongchun Lu, Liying Kang
TL;DR
The paper develops a unified α-spectral Turán theory by studying the α-spectral radius λ_α(G) with A_α(G)=αD(G)+(1-α)A(G). It proves an α-spectral Erdős-Stone-Simonovits-type asymptotic ex for χ(F)=r+1≥3 and 0≤α≤1-1/r, and extends color-critical results by showing that, for large n and 0≤α<1-1/r, the Turán graph T_{n,r} uniquely maximizes λ_α among F-free graphs, with a corresponding framework of auxiliary lemmas and degree-stability. The work unifies existing adjacency and signless Laplacian spectral extremal results and provides explicit equality cases, broadening the scope of spectral Turán-type problems. Overall, it offers a rigorous α-parameterized bridge between Turán-type extremal graph theory and spectral graph theory.
Abstract
For $0 \leq α< 1$, the $α$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$, respectively. A graph is called color-critical if it contains an edge whose deletion reduces its chromatic number. The celebrated Erdős-Stone-Simonovits theorem asserts that $ \mathrm{ex}(n,\mathcal{F})=\left(1-\frac{1}{χ(\mathcal{F})-1}+o(1)\right)\frac{n^2}{2},$ where $χ(\mathcal{F})$ is the chromatic number of $\mathcal{F}$. Nikiforov and Zheng et al. established the adjacency spectral and signless Laplacian spectral versions of this theorem, respectively. In this paper, we present the $α$-spectral version of this theorem, which unifies the aforementioned results. Furthermore, we characterize the $α$-spectral extremal graphs for color-critical graphs, thereby extending the existing results on adjacency spectral and signless Laplacian spectral extremal graphs for such graphs.