On Fiedler- and Parter-vertices of acyclic matrices

Let A be a real symmetric matrix and let λ be a real number. The algebraic multiplicity of λ as an eigenvalue of A is denoted by mA(λ), and the principal submatrix of A obtained by deleting row and column i from A is denoted by A(i). If mA(i)(λ)⩾mA(λ) (resp. mA(i)(λ)>mA(λ)), then index i is said to be a Fiedler-vertex (resp. a Parter-vertex) of A for λ. In this paper we provide geometric characterizations of Fiedler- and Parter-vertices of acyclic matrices, and give a geometric proof for the Parter–Wiener theorem in [C.R. Johnson, A. Leal Duarte, C.M. Saiago, The Parter–Wiener theorem: refinement and generalization, SIAM J. Matrix Anal. Appl. 25 (2003) 352–361]. Furthermore, we describe a structure of an acyclic matrix in terms of Fiedler- and Parter-vertices which enables us to construct an acyclic matrix of a desired form according to the locations of Fiedler- and Parter-vertices.