dc.contributor.author Fiol Mora, Miquel Àngel dc.date.accessioned 2011-05-12T12:21:29Z dc.date.available 2011-05-12T12:21:29Z dc.date.issued 2011 dc.identifier.citation Fiol Mora, Miquel Àngel. Algebraic characterizations of bipartite distance-regular graphs. A: International Workshop on Optimal Networks Topologies. "Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010". Barcelona: Iniciativa Digital Politècnica, 2011, p. 253-264. dc.identifier.isbn 978-84-7653-565-3 dc.identifier.uri http://hdl.handle.net/2099/10389 dc.description.abstract Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs. Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local multiplicities (entries of the idempotents $E_i$ or eigenprojectors), the predistance polynomials, etc. For instance, it is known that a graph G, with eigenvalues $λ_0$ > $λ_1$ > · · · > $λ_d$ and diameter D = d, is distance-regular if and only if its idempotents $E_1$ and $E_d$ belong to the vector space D spanned by its distance matrices I,A,$A_2$, . . .$A_d$. In contrast with this, for the same result to be true in the case of bipartite graphs, only $E_1$ ∈ D need to be required. dc.format.extent 12 p. dc.language.iso eng dc.publisher Iniciativa Digital Politècnica dc.relation.ispartof International Workshop on Optimal Networks Topologies dc.relation.uri http://hdl.handle.net/2099.2/1750 dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs dc.subject.lcsh Bipartite graphs dc.subject.lcsh Eigenvalues dc.subject.lcsh Combinatorial analysis dc.title Algebraic characterizations of bipartite distance-regular graphs dc.type Conference report dc.subject.lemac Grafs, Teoria de dc.subject.lemac Valors propis dc.subject.lemac Anàlisi combinatòria dc.description.peerreviewed Peer Reviewed dc.subject.ams Classificació AMS::05 Combinatorics::05C Graph theory dc.rights.access Open Access local.citation.author Fiol Mora, Miquel Àngel local.citation.contributor International Workshop on Optimal Networks Topologies local.citation.pubplace Barcelona local.citation.publicationName Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010 local.citation.startingPage 253 local.citation.endingPage 264
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• 3er. 2010 [27]
Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, 9-11 June 2010