四叶图距离矩阵2个最大特征值和的变化

Variation of sum of two largest eigenvalues of the distance matrices of four-leaf graph

为了能够在任何情况下准确得到四叶图在2种图变换下距离特征值的极值,运用行列式的性质、韦达定理及不等式的放缩,给出了四叶图的2种图变换及上述问题的结果。首先分别给出变换前后3种四叶图距离矩阵、距离拉普拉斯矩阵及距离无符号拉普拉斯矩阵,利用行列式的性质计算得出其特征多项式,由韦达定理判断出3种距离特征多项式正负根的个数,通过不等式的放缩估计出特征值的范围,从而求出2个最大特征值和的范围;其次对变化前后四叶图的3种距离矩阵2个最大特征值的和进行比较。结果显示,四叶图在经过2种变换后2个最大特征值的和是增加的。所得结果为特殊图类距离特征值极值问题提供了研究方法,对分子稳定性问题的研究具有一定的借鉴价值。

In order to accurately obtain the extremum of the distance eigenvalues of fowr-leaf graphs under tow graph transformations in any case,two graph transformations of four-leaf graphs and the results of the above problems were given by using the properties of the determinant,the Vieta theorem and the reduction of inequality.Firstly,the distance matrices,the distance Laplacian matrices and the distance signless Laplacian matrices of three kinds of four-leaf graphs before and after the transformation were given.The characteristic polynomials were obtained by using the properties of the determinant.The number of positive and negative roots of three kinds of distance characteristic polynomials was determined by the Vieta theorem.The range of eigenvalues was estimated by the reduction of inequality.Thus,the range of the sum of the two maximum eigenvalues was obtained.Finally,the two maximum eigenvalues of three kinds of distance matrices of four-leaf graphs before and after the transformations were compared.The comparison shows that the sum of the two maximum eigenvalues of four-leaf graph increases after two transformations.The results provide a research method for the extremum problems of distance eigenvalues of special graphs,and have certain reference value for the research of moleculer stability.