The real symmetric matrices with a P-set of maximum size and their associated graphs

The concept of P-set of real symmetric matrices is proposed based on the multiplicity of eigenvalues of matrices and Cauchy interlacing theory. Suppose that is a real symmetric matrix of order . Let be the multiplicity of eigenvalue 0 of , and let be the principal submatrix of obtained from by deleting the rows and columns, where is a nonempty subset of . In particular, when , we say S is a P-set of . Let be the maximum size among the P-sets of . Kim and Shader proved that every real symmetric matrix of order has at more elements, i.e., . Du and Fonseca first focused on the acyclic matrices (i.e., the matrices whose associated graphs are trees), investigated the acyclic matrices of order n satisfied , and completely characterized the associated graphs (trees) of .This paper extended the research from acyclic matrices to all real symmetric matrices, investigated the real symmetric matrices of order satisfied , presented their properties, and characterized the associated graphs when is even, conjectured the associated graphs when is odd, which improved the results on acyclic matrices.