谱Turán条件下三角形数量的稳定性估计

Stability Estimates for Triangle Counts under the Spectral Turán Condition

摘要: 在谱Turán理论中，图的谱半径被视为刻画图整体稠密程度的重要参数。本文研究谱Turán临界条件下三角形数量的稳定性问题。设$G$ 为一个$n$ 阶简单图，其谱半径满足$\lambda \left(G\right)\ge \lambda \left(K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }^{+1}\right)$ ，其中$K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }^{+1}$ 表示由完全二部$K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }$ 图在大小为$\lceil \frac{n}{2}\rceil$ 的部集中添加一条独立边得到的图。我们证明：若$G$ 至少含有一个三角形，且不存在一个顶点同时属于所有三角形，则当$n$ 充分大时，图$G$ 中三角形的总数具有线性下界，即存在一个与$n$ 无关的常数$c\ge 2$ ，使得三角形数不少于$n-c$ 。该结果可视为对已有谱条件下三角形计数结果的稳定性加强。证明中结合了三角形计数不等式、二部图逼近方法以及极值图的谱半径比较等方法。

Abstract: In spectral Turán theory, the spectral radius of a graph is regarded as an important parameter for measuring its global density. In this paper, we study the stability of triangle counts under the spectral Turán threshold. Let $G$ be a simple graph on $n$ vertices whose spectral radius satisfies $\lambda \left(G\right)\ge \lambda \left(K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }^{+1}\right)$ , where $K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }^{+1}$ denotes the graph obtained from the complete bipartite graph $K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }$ by adding one edge inside the larger part. We prove that if $G$ contains at least one triangle and no single vertex lies in all triangles, then for sufficiently large $n$ , the total number of triangles in $G$ admits a linear lower bound. More precisely, there exists a constant $c\ge 2$ independent of $n$ , such that the number of triangles in $G$ is at least $n-c$ . This result provides a stability strengthening of previous spectral triangle-counting results. The proof combines triangle counting inequalities, bipartite approximation techniques, and comparisons of spectral radii of extremal graphs.