Let G be a connected simple graph with characteristic polynomial $P_G(x)$. The irreducibility of $P_G(x)$ over rational numbers $\mathbb{Q}$ has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths $P_{2^{n+1}-2}$ $(n\ge 1)$ to certain vertices; union and join $K_1$ alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.

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关于具有不可约特征多项式的图形的注意事项

令G为具有特征多项式的连通简单图${磷}_G(X)$. 的不可约性${磷}_G(X)$ 超过有理数 $\mathbb{问}$与图的自同构群、重构和可控性有密切关系。在本文中，我们推导出了三种方法来通过附加路径来构造具有不可约特征多项式的图${磷}_{2^{n+1}-2}$ $(n\ge 1)$到某些顶点；联合和加入${钾}_1$交替和电晕。这些方法基于爱森斯坦的标准和领域扩展。还提供了具体的例子来说明我们的结果。