The local spectra of regular line graphs

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 is an element of V, the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue lambda 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. (C) 2009 Elsevier B.V. All rights reserved.