The local spectrum of a graph G=(V,E), constituted by the standard eigenvalues of G 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@?V, the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue @l of G 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 LG in terms of the local spectrum of the regular graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LG.
['Miguel Angel Fiol', 'Margarida Mitjana']
The local spectra of regular line graphs