摘要或動機

Internet search technology is a pervasively used utility that relies on techniques from the _eld of spectral graph theory. We present a novel spectral approach to investigate an existing problem: the critical group of the line graph has been characterized for regular nonbipartite graphs, but the general regular bipartite case remains open. Because of the ine_ectiveness of previous techniques in regular bipartite graphs, our approach provides a new perspective and aims to obtain the relationship between the spectra of the Laplacians of the graph G and its line graph bG. We obtain a theorem for the spectra of all regular bipartite graphs and demonstrate its e_ectiveness by completely characterizing the previously unknown critical group for a particular class of regular bipartite graphs, the incidence graphs of _nite projective planes with square order. This critical group is found to be Z2_(Z2q+2)q3􀀀1_(Zq2+q+1)q2+q􀀀1; where q is the order of the _nite projective plane.