对于边色临界图，非$r$部谱极值图是边极值图
A reduction principle for non-$r$-partite spectral extremal problems, with a complete multipartite classification
AI总结：
本文证明在特定条件下，非$r$部$F$-free图中达到邻接谱半径最大的图也达到边数最大，并应用于完全多部图$K_{1,1,t_3,\ldots,t_{r+1}}$。
AI中文摘要：
如果一个图的色数超过$r$，则称其为非$r$部图。对于边色临界图$F$且$\chi(F)=r+1$，令$\mathrm{ex}_{r+1,\rho}(n,F)$为$n$阶非$r$部$F$-free图中邻接谱半径的最大值，并令$\mathrm{EX}_{r+1,\rho}(n,F)$和$\mathrm{EX}_{r+1}(n,F)$分别为达到该最大谱半径和最大边数$\mathrm{ex}_{r+1}(n,F)$的图族。Fang和Zhai猜想：对于每个这样的$F$和所有足够大的$n$，有$\mathrm{EX}_{r+1,\rho}(n,F)\subseteq\mathrm{EX}_{r+1}(n,F)$。本文在假设$\mathrm{ex}_{r+1}(n,F)=|E(T_{n,r})|-\lfloor n/r\rfloor+O(1)$（其中$T_{n,r}$是Turán图）以及$F$的子分解族的一个结构条件下，证明了该包含关系。作为主要应用，对于$F=K_{1,1,t_3,\ldots,t_{r+1}}$且$t_3,\ldots,t_{r+1}\ge 2$，我们证明对所有足够大的$n$有\[ \mathrm{ex}_{r+1}(n,F)=|E(T_{n,r})|-\Bigl\lfloor\frac nr\Bigr\rfloor+2(t_{\min}-1), \qquad t_{\min}:=\min\{t_3,\ldots,t_{r+1}\}, \]并推出$\mathrm{EX}_{r+1,\rho}(n,F)\subseteq\mathrm{EX}_{r+1}(n,F)$。
英文摘要：
A graph is non-$r$-partite if its chromatic number exceeds $r$. For an edge-color-critical graph $F$ with $χ(F)=r+1$, let $\mathrm{ex}_{r+1,ρ}(n,F)$ be the maximum adjacency spectral radius among non-$r$-partite $F$-free graphs of order $n$, and let $\mathrm{EX}_{r+1,ρ}(n,F)$ and $\mathrm{EX}_{r+1}(n,F)$ be the families of such graphs attaining, respectively, this maximum spectral radius and the maximum number of edges $\mathrm{ex}_{r+1}(n,F)$. Fang and Lin conjectured that $\mathrm{EX}_{r+1,ρ}(n,F)\subseteq\mathrm{EX}_{r+1}(n,F)$ for every such $F$ and all large $n$. We prove a reduction principle: if $F$ is \emph{$s$-embeddable} and $\mathrm{ex}_{r+1}(n,F)=|E(T_{n,r})|-\lfloor n/r\rfloor+2(s-1)$, where $T_{n,r}$ is the Turán graph, then the inclusion holds and, moreover, the spectral extremal graph is unique. The reduction replaces the spectral problem by an edge-counting one, and its proof rests on a direct comparison of secular functions together with a second-order residual refinement of the Rayleigh principle. We then determine the spectral extremal graphs for all edge-color-critical complete multipartite forbidden graphs. For $F=K_{1,1,t_3,\ldots,t_{r+1}}$ with $t_3,\ldots,t_{r+1}\ge 2$ we show \[ \mathrm{ex}_{r+1}(n,F)=|E(T_{n,r})|-\Bigl\lfloor\frac nr\Bigr\rfloor+2(t_{\min}-1), \qquad t_{\min}:=\min\{t_3,\ldots,t_{r+1}\}, \] for all sufficiently large $n$, and we identify the unique spectral extremal graph; in particular $\mathrm{EX}_{r+1,ρ}(n,F)\subseteq\mathrm{EX}_{r+1}(n,F)$. The endpoint $t_3=1$ lies outside the embeddability framework and is treated by a separate argument: for $F=K_{1,1,1,t_4,\ldots,t_{r+1}}$ with $r\ge3$, the unique non-$r$-partite spectral extremal graph is $Y_r(n)$, obtained through a saturation reduction followed by the spectral refinement of Turán's theorem. The complete graph $K_{r+1}$ and the complete split graph $B_{r,q}$
