Matrix power inequalities and the number of walks in graphs

We unify and generalize several inequalities for the number  wkwk of walks of length  kk in graphs, and for the entry sum of matrix powers.First, we present a weighted sandwich theorem for Hermitian matrices which generalizes a matrix theorem by Marcus and Newman and which further generalizes our former unification of inequalities for the number of walks in undirected graphs by Lagarias et al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weighting of the indices (vertices) which allows to apply the theorem to index (vertex) subsets (i.e., inequalities considering the number wk(S,S)wk(S,S) of walks of length  kk that start at a vertex of a given vertex subset  SS and that end within the same subset). We also deduce a stronger variation of the sandwich theorem for the case of positive-semidefinite Hermitian matrices which generalizes another inequality of Marcus and Newman.Further, we show a similar theorem for nonnegative symmetric matrices which is another unification and generalization of inequalities for walk numbers by Erdős and Simonovits, by Dress and Gutman, and by Ilić and Stevanović.In the last part, we generalize lower bounds for the spectral radius of adjacency matrices and upper bounds for the energy of graphs.