  Protein-protein interaction (PPI) networks are the backbone of all processes
in living cells. In this work we relate conservation, essentiality and
functional repertoire of a gene to the connectivity $k$ (i.e., the number of
interaction links) of the corresponding protein in the PPI network. On a set of
42 bacterial genomes of different sizes, and with reasonably separated
evolutionary trajectories, we investigate three issues: i) whether the
distribution of connectivities changes between PPI subnetworks of essential and
nonessential genes; ii) how gene conservation, measured both by the
evolutionary retention index (ERI) and by evolutionary pressures, is related to
the the connectivity of the corresponding protein; iii) how PPI connectivities
are modulated by evolutionary and functional relationships, as represented by
the Clusters of Orthologous Genes (COGs). We show that conservation,
essentiality and functional specialisation of genes constrain the connectivity
of the corresponding proteins in bacterial PPI networks. In particular, we
isolate a core of highly connected proteins (with connectivities $k\ge40$),
which is ubiquitous among the species considered here -- though mostly visible
in the degree distributions of bacteria with small genomes (less than 1000
genes). The genes that belong to this highly connected core are conserved,
essential and, in most cases, belong to the COG cluster J, related to ribosomal
functions and to the processing of genetic information.
