The local spectra of regular line graphs

The local spectrum of a graph G=(V,E)G=(V,E), constituted by the standard eigenvalues of GG and their local multiplicities, plays a similar role as the global spectrum when the graph is “seen” from a given vertex. Thus, for each vertex i∈Vi∈V, the ii-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λλ of GG is the sum, extended to all vertices, of its local multiplicities.In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LGLG in terms of the local spectrum of the regular graph GG it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LGLG.