Dual concepts of almost distance-regularity and the spectral excess theorem

TL;DR

This paper introduces dual concepts of almost distance-regularity in graphs, characterizes specific subclasses using spectral properties, and generalizes the spectral excess theorem to deepen understanding of distance-regular graphs.

Contribution

It proposes two new dual notions of almost distance-regularity and extends the spectral excess theorem to these concepts, enhancing the theoretical framework.

Findings

Characterization of m-partially distance-regular graphs via spectra

Characterization of j-punctually eigenspace distance-regular graphs via spectra

Generalization of the spectral excess theorem to dual concepts

Abstract

Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity, thus giving a better understanding of the properties of distance-regular graphs. More precisely, we characterize $m$-partially distance-regular graphs and $j$-punctually eigenspace distance-regular graphs by using their spectra. Our results can also be seen as a generalization of the so-called spectral excess theorem for distance-regular graphs, and they lead to a dual version of it.