On the evaluation of the characteristic polynomial via symmetric function theory

Abstract

The use of power sum symmetric functions leads to Newton's identities, which relate the traces of various powers ofA, the adjacency matrix of a graph, and the coefficients of the characteristic polynomials. While it is possible to solve Newton's identities and generate the coefficients by recursion or, alternatively, to derive them by sequential manipulations (yielding the explicit formulas), we show how the results can be expressed using a combinatorial approach and relate the evaluation of the coefficients to selected Young diagrams.