On the algebraic theory of pseudo-distance-regularity around a set Original Research Article

Let Γ be a connected graph with vertex set V, adjacency matrix A, positive eigenvector ν and corresponding eigenvalue λ0. A natural generalization of distance-regularity around a vertex subset Csubset ofV, 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 uset membership, variantV a weight which equals the corresponding entry νu of ν 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 λ0) if and only if Γ 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.