Centralities of a network and its line graph: An analytical comparison by means of their irregularity

Given a network G, it is known that the Bonacich centrality of the bipartite graph B(G) associated with G can be obtained in terms of the centralities of the line graph L(G) associated with G and the centrality of the network G+gr (whose adjacency matrix is obtained by adding to the adjacency matrix A(G) the diagonal matrix D=bij, where bii is the degree of node i in G) and conversely. In this contribution, we use the centrality of G to estimate the centrality of G+gr and show that the error committed is bounded by some measure of the irregularity of G. This estimate gives an analytical comparison of the eigenvector centrality of G with the centrality of L(G) in terms of some irregularity measure of G.