On the multiplicity of the eigenvalues of a graph (Q879222)

Given a graph \(G\), the characteristic polynomial of \(G\) is \(\varphi (t)=\varphi_{A(G)}(t)=\det (tI_n-A)\), where \(A\) is the adjacency matrix of \(G\). The multiplicity layered (ML)-decomposition \(\varphi (t)=q_1(t)q_2(t)^2\dots q_m(t)^m\) is considered, where each \(q_i(t)\) is an integral polynomial and the roots of \(\varphi (t)\) with multiplicity \(j\) are exactly the roots of \(q_j(t)\). An algorithm to construct the polynomials \(q_i(t)\) is given as well as the description of the relation of their coefficients with other invariants of \(G\). In particular, new bounds for the energy \(E(G)=\sum_{i=1}^{n}| \lambda _i| \) are obtained, where \(\lambda _1, \lambda _1,\dots ,\lambda _n\) are the eigenvalues of \(G\) (with multiplicity).