For $n\ge 2$, let ${\mathbb{S}}_n^{\ast }=\left\{\right(0,n,0),(0,n-1,1),(1,n-1,0),(n,0,0),(n-1,0,1),(n-1,1,0\left)\right\}$. A zero-nonzero pattern $\mathcal{A}$ of order n allows ${\mathbb{S}}_n^{\ast }$ if ${\mathbb{S}}_n^{\ast }\subseteq \mathrm{i}\left(\mathcal{A}\right)$, the inertia set of $\mathcal{A}$, and requires ${\mathbb{S}}_n^{\ast }$ if ${\mathbb{S}}_n^{\ast }=\mathrm{i}\left(\mathcal{A}\right)$. The study of zero-nonzero patterns allowing ${\mathbb{S}}_n^{\ast }$ was introduced by Berliner et al. [Inertia sets allowed by matrix patterns. Electron J Linear Algebra. 2018;34:343–355]. In this paper, we give characterizations of irreducible zero-nonzero path patterns and double-star patterns that allow ${\mathbb{S}}_n^{\ast }$. Moreover, all irreducible zero-nonzero tree patterns of order 5 and 6 that allow ${\mathbb{S}}_n^{\ast }$ are also characterized.

为了$n\ge \mathrm{2个}$， 让${\mathbb{小号}}_n^{＊}=\left\{\right(0,n,0),(0,n-\mathrm{1个},\mathrm{1个}),(\mathrm{1个},n-\mathrm{1个},0),(n,0,0),(n-\mathrm{1个},0,\mathrm{1个}),(n-\mathrm{1个},\mathrm{1个},0\left)\right\}$. 零-非零模式$\mathcal{A}$n阶允许${\mathbb{小号}}_n^{＊}$如果${\mathbb{小号}}_n^{＊}\subseteq \mathrm{我}\left(\mathcal{A}\right)$, 惯性集$\mathcal{A}$, 并要求${\mathbb{小号}}_n^{＊}$如果${\mathbb{小号}}_n^{＊}=\mathrm{我}\left(\mathcal{A}\right)$. 零-非零模式的研究允许${\mathbb{小号}}_n^{＊}$由 Berliner 等人介绍。[矩阵模式允许的惯性集。电子 J 线性代数。2018 年；34：343–355]。在本文中，我们给出了不可约零-非零路径模式和双星模式的特征，这些模式允许${\mathbb{小号}}_n^{＊}$. 此外，所有 5 阶和 6 阶的不可约零-非零树模式允许${\mathbb{小号}}_n^{＊}$也有特点。