图的拉普拉斯矩阵及其特征值已被应用于数学研究的许多领域,是图在同构意义下的不变量,因此拉普拉斯特征值就决定着图的结构（在同构意义下）,由于在拉普拉斯矩阵中含有图的顶点度的信息, 所以图的拉普拉斯特征值与图的很多不变量有着密切的联系. 本文所研究的均是对含独立圈的图而言的,具体来说主要研究了拉普拉斯特征值2和3的存在性及其重数与图的结构之间的关系,主要结果分为以下几点:(1)连通图中整数特征值的存在性.(2)对含完美匹配的残缺太阳图而言,2是其拉普拉斯特征值,并且对应的特征向量在每一个顶点处的分量均非零,且绝对值相等.(3)在含独立圈图中,拉普拉斯特征值2的存在性及其重数.(4)在辐射图中,拉普拉斯特征值3的存在性及其重数.

The Laplacian matrix and its eigenvalues have been applied in many areasof mathematical research, it’s the invariant under the isomorphism meaning, sothe laplacian eigenvalues determines the structure of graphs. Because the lapla-cian matrix contains the degree information of vertices of graphs, so the laplacianeigenvalues have closer links with other invariant of graph. In this paper, we dis-cuss the existence of laplacian eigenvalue 2 and 3 and the relationship between theexistence and the structure of graphs.Specifically, it’s the following main research:(1) The existence of integer eigenvalue of the connected graphs;(2) For the broken sun graph containing a perfect matching, 2 is one of thelaplacian eigenvalues and every component of the corresponding eigenvector isnot zero, further the absolute value of the component is equal;(3) For the independent circle graphs , the existence and multiplicities oflaplacian eigenvalue 2;(4) For the radiation graphs with specific structure, the existence and multi-plicities of laplacian eigenvalue 3;