  A unified framework for the Expander Mixing Lemma for irregular graphs using
adjacency eigenvalues is presented, as well as two new versions of it. While
the existing Expander Mixing Lemmas for irregular graphs make use of the notion
of volume (the sum of degrees within a vertex set), we instead propose to use
the Perron eigenvector entries as vertex weights, which is a way to regularise
the graph. This provides a new application of weight partitions of graphs. The
new Expander Mixing Lemma versions are then applied to obtain several
eigenvalue bounds for NP-hard parameters such as the zero forcing number, the
vertex integrity and the routing number of a graph.
