Inverse Spectral Problems Of Two Types Of Acyclic Matrices

Spectral Graph Theory originally focused on specific matrices,such as the adjacency matrix or the Laplacian matrix,whose entries are determined by the graph,with the goal of obtaining information about the graph from the matrices.In contrast,the Inverse Eigenvalue Problem of a Graph seeks to determine information about the possible spectra of the real symmetric matrices whose pattern of nonzero entries is described by a given graph.This paper studies the inverse eigenvalue problems of two types of special matrices described by a given graph,which are the double starlike matrix extended from the matrix whose graph is broom and the arrow matrix.Regarding the first type of double starlike matrix,which is generalized by the matrix whose graph is a broom,firstly,given two types of different eigen data,it is reconstructed by solving the linear equations and the recursive relationship of the eigen polynomials of the principal submatrix of distinct order.For the required matrix,the necessary and sufficient conditions and algorithms for the unique solution of the inverse eigenvalue problem of this type of matrix are given.Finally,the effectiveness of the algorithm is verified by two specific numerical simulation examples.The second kind of graph is the arrow matrix of the centipede.According to the eigen pair and the maximum eigenvalue of the given matrix,the matrix is constructed by using the recursive relationship between the characteristic polynomial of each order principal submatrix,and the expression of the necessary and sufficient condition solution is specified,so that the matrix has a unique solution.Finally,the accuracy of the result is verified by numerical simulation examples.