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 centra...
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.
Citation
Fiol, M. A.; Garriga, E. On the algebraic theory of pseudo-distance-regulariry around a set. "Linear algebra and its applications", Setembre 1999, vol. 298, p. 115-141.