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The local spectra of regular line graphs
Fiol Mora, Miquel Àngel; Mitjana Riera, Margarida
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions
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 λ 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.
2012-05-10
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
Eigenvalues
Graph theory
Graph spectrum
Eigenvalue
Local multiplicity
Line graph
Valors propis
Grafs, Teoria de
/Classificació AMS/05 Combinatorics/05C Graph theory
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