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

Abstract

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.

Keywords

• graph theory; four-leaf graph; distance matrix; eigenvalue; graph transformation