  We investigate the effect of the node degree and energy $E$ on the electronic
wave function for regular and irregular structures, namely, regular lattices,
disordered percolation clusters, and complex networks. We evaluate the
dependence of the quantum probability for each site on its degree. For
bi-regular structures, we prove analytically that the probability $P_k(E)$ of
finding the particle on any site with $k$ neighbors is independent of $E$. For
more general structures, the dependency of $P_k(E)$ on $E$ is discussed by
taking into account exact results on a one-dimensional semi-regular chain:
$P_k(E)$ is large for small values of $E$ when $k$ is also small, and its
maximum values shift towards large values of $|E|$ with increasing $k$.
Numerical evaluations of $P_k(E)$ for two different types of percolation
clusters and the Apollonian network suggest that this feature might be
generally valid
