Newman’s measure for (dis)assortativity, the linear degree correlation
coefficient \(\rho _{D}\), is reformulated in terms of the total number N_{k}
of walks in the graph with k hops. This reformulation allows us to derive
a new formula from which a degree-preserving rewiring algorithm is deduced,
that, in each rewiring step, either increases or decreases \(\rho _{D}\)
conform our desired objective. Spectral metrics (eigenvalues of
graph-related matrices), especially, the largest eigenvalue \(\lambda _{1}\)
of the adjacency matrix and the algebraic connectivity \(\mu _{N-1}\)
(second-smallest eigenvalue of the Laplacian) are powerful characterizers of
dynamic processes on networks such as virus spreading and synchronization
processes. We present various lower bounds for the largest eigenvalue \(\lambda _{1}\) of the
adjacency matrix and we show, apart from some classes
of graphs such as regular graphs or bipartite graphs, that the lower bounds
for \(\lambda _{1}\) increase with \(\rho _{D}\). A new upper bound for the
algebraic connectivity \(\mu _{N-1}\) decreases with \(\rho _{D}\). Applying the
degree-preserving rewiring algorithm to various real-world networks
illustrates that (a) assortative degree-preserving rewiring increases \(\lambda _{1}\), but decreases
\(\mu _{N-1}\), even leading to disconnectivity
of the networks in many disjoint clusters and that (b) disassortative
degree-preserving rewiring decreases \(\lambda _{1}\), but increases the
algebraic connectivity, at least in the initial rewirings.