We first prove a formula which relates the characteristic polynomial of a matrix (or of a weighted graph), and some invariants obtained from its principal submatrices (resp. vertex deleted subgraphs). Consequently, we express the spectral radius of the observed objects in the form of power series. In particular, as is relevant for the spectral graph theory, we reveal the relationship between spectral radius of a simple graph and its combinatorial structure by counting certain walks in any of its vertex deleted subgraphs. Some computational results are also included in the paper.