On the algebraic theory of pseudo-distance-regularity around a set
Copyright policy )On the algebraic theory of pseudo-distance-regularity around a set
Let C be a connected graph with vertex set V, adjacency matrix A, positive eigenvector and corresponding eigenvalue 0. A natural generalization of distance-regularity around a vertex subset C V , which makes sense even with non-regular graphs, is studied. This new concept is called pseudo-distance-regularity, and its definition is based on giving to each vertex u 2 V a weight which equals the corresponding entry u of and “regularizes” the graph. This approach reveals a kind of central symmetry which, in fact, is an inherent property of all kinds of distance-regularity, because of the distance partition of V they come from. We come across such a concept via an orthogonal sequence of polynomials, constructed from the “local spectrum” of C, called the adjacency polynomials because their definition strongly relies on the adjacency matrix A. In particular, it is shown that C is “tight” (that is, the corresponding adjacency polynomials attain their maxima at 0) if and only if C is pseudo-distance-regular around C. As an application, some new spectral characterizations of distance-regularity around a set and completely regular codes are given.
Peer Reviewed
adjacency matrix, Orthogonal polynomials, Local spectrum, orthogonal sequence of polynomials, Grafs, eigenvalue, Discrete Mathematics and Combinatorics, Orthogonal polynomials (combinatorics), completely regular codes, Numerical Analysis, Algebra and Number Theory, Teoria de, Graphs and linear algebra (matrices, eigenvalues, etc.), Grafs, Teoria de, 05C Graph theory, Completely regular code, Distance-regular graph, characterizations, Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs, distance-regularity, eigenvector, Graph theory, 05C38, Association schemes, strongly regular graphs, Geometry and Topology, 05C50, Paths and cycles, :Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs [Àrees temàtiques de la UPC]
adjacency matrix, Orthogonal polynomials, Local spectrum, orthogonal sequence of polynomials, Grafs, eigenvalue, Discrete Mathematics and Combinatorics, Orthogonal polynomials (combinatorics), completely regular codes, Numerical Analysis, Algebra and Number Theory, Teoria de, Graphs and linear algebra (matrices, eigenvalues, etc.), Grafs, Teoria de, 05C Graph theory, Completely regular code, Distance-regular graph, characterizations, Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs, distance-regularity, eigenvector, Graph theory, 05C38, Association schemes, strongly regular graphs, Geometry and Topology, 05C50, Paths and cycles, :Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs [Àrees temàtiques de la UPC]
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).26 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 10% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10%
Citations provided by BIP!Popularity provided by BIP!