on the inverse eigenvalue problem for a specific symmetric matrix

The aim of the current paper is to study a partially described inverse eigenvalue problem of  a specific symmetric  matrix, and prove some properties of such matrix. the problem includes the construction of the matrix by  the  minimal eigenvalue of all  leading principal submatrices  and eigenpair $(lambda_2^{(n)},x)$ such that $ lambda_2^{(n)}$ is the maximal eigenvalue of the required matrix. we investigate  conditions for the solvability of the problem, and finally an algorithm and  its numerical results are presented.