禁止圈与θ图条件下达到最大无符号拉普拉斯谱半径的图的刻画

中文摘要 AI 辅助

谱Turán型问题探讨的是：禁止某些指定子图的存在，如何约束与图相关联的矩阵的谱半径。给定一个图族$\mathcal{F}$，若一个图不包含$\mathcal{F}$中任何成员作为子图，则称该图为$\mathcal{F}$-自由图。θ图$\theta(l_1,\ldots,l_k)$由$k$条内部不相交、长度分别为$l_1,\ldots,l_k$的路径组成，这些路径共享两个公共端点。本文研究了无符号拉普拉斯谱半径的两个谱Turán型极值问题。首先，在所有无悬挂点的、固定阶数的$\{C_3,C_4\}$-自由图中，我们确定了最大无符号拉普拉斯谱半径，并唯一刻画了达到该最大值的极值图。极值结构表现出奇偶性现象：奇数阶和偶数阶分别产生两个不同的图族。这些结果特别地改进了Liu和Wang（2026）近期给出的这一类的通用上界。其次，对于无悬挂点、固定边数且边数模3余1和余2的所有$\{\theta(1,2,2),\theta(1,2,3)\}$-自由图，我们获得了相应的极值结果，同样在两种情形下得到了唯一但结构不同的最大化图。结合Liu和Wang（2026）先前对边数模3余0情形的已知结果，这完成了模3所有三个同余类下的固定边数问题。

英文摘要

Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs $\mathcal{F}$, a graph is called $\mathcal{F}$-free if it contains no member of $\mathcal{F}$ as a subgraph. The theta graph $θ(l_1,\ldots,l_k)$ consists of $k$ internally disjoint paths of lengths $l_1,\ldots,l_k$ with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all $\{C_3,C_4\}$-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all $\{θ(1,2,2),θ(1,2,3)\}$-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.

发表机构

• Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）

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