The generalized $H(n)$ Hirsch index of order $n$ has been recently introduced and shown to interpolate between the degree and the $K$-core centrality in networks. We provide a detailed analytical characterization of the properties of sets of nodes having the same $H(n)$, within the annealed network approximation. The connection between the Hirsch indices and the degree is highlighted. Numerical tests in synthetic uncorrelated networks and real-world correlated ones validate the findings. We also test the use of the Hirsch index for the identification of influential spreaders in networks, finding that it is in general outperformed by the recently introduced Non-Backtracking centrality.
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