多项式是代数学中最基本的研究对象之一，而不可约多项式作为一种重要的多项

多项式是代数学中最基本的研究对象之一，而不可约多项式作为一种重要的多项式，与所有多项式相关，它在多项式环中有类似于素数在整数环中的地位。现在可约性的概念已经渗透在数学的各个分支，在不同的分支中都有不同的表现形式。因此研究多项式的不可约性有着非常重要的理论意义和应用价值，可以帮助我们解决许多有用的问题。<br>有理数域上多项式不可约性的研究由来已久，并作为线性代数和高等代数的基础知识进行了详细论述，现在，关于有理数域上的不可约多项式的研究仍备受关注。图的特征多项式 作为一种特殊的多项式，关于它的不可约性的研究对于判断有理系数多项式是否可约具有一定的借鉴意义.<br>关于图的特征多项式的不可约性的判定，首先确定首系数，看是否可以使用模m约化判别法，如果符合条件，用该方法进行判定；不管首系数如何，都可以用Eisenstein判别法来对图的特征多项式的不可约性进行判定；然后进行科学性地探究，看是否可以找到其他方法来解决图的特征多项式的不可约性问题。最后根据Mathematica软件的基本操作，找出七个点图的特征多项式，判断这些多项式的可约性，得出结论。

其中一个著名的是Shibam Hadramuot，一个高大的，密集，泥砖建筑的奇妙地点，16世纪的围墙城市希巴姆，被誉为世界上第一个高层公寓楼的家，建于公元300年。

Polynomials are one of the most basic research objects in algebra. Irreducible polynomials, as an important kind of polynomials, are related to all polynomials. They have the similar position in polynomial rings as prime numbers in integer rings. Now the concept of reducibility has penetrated into every branch of mathematics, and it has different forms in different branches. So it is very important to study the irreducibility of polynomials, which can help us to solve many useful problems.<br>The irreducibility of polynomials in rational number field has been studied for a long time. As the basic knowledge of linear algebra and advanced algebra, the research on irreducible polynomials in rational number field is still concerned. As a special kind of polynomials, the study of irreducibility of the characteristic polynomials of graphs has certain reference significance for judging whether the rational coefficient polynomials are irreducible<br>To determine the irreducibility of the characteristic polynomials of a graph, first determine the first coefficient to see whether the modular m-reduction method can be used. If the conditions are met, use this method to determine; regardless of the first coefficient, use Eisenstein method to determine the irreducibility of the characteristic polynomials of a graph; then explore scientifically to see whether other methods can be found To solve the irreducibility of characteristic polynomials of graphs. Finally, according to the basic operation of Mathematica software, we find out the characteristic polynomials of seven point graphs, judge the reducibility of these polynomials, and draw a conclusion.<br>