On almost distance-regular graphs
Cristina Dalf\'o, Edwin R. van Dam, Miquel Angel Fiol, Ernest Garriga,, Bram L. Gorissen

TL;DR
This paper explores various generalizations of distance-regular graphs, introducing concepts like almost distance-regularity, m-walk-regularity, and punctual regularities, using spectral methods and examples from symmetric cubic graphs.
Contribution
It defines and relates new concepts of almost distance-regularity, providing characterizations and examples, and poses open problems about when these approximate properties become exact.
Findings
Introduced m-walk-regularity and m-partial distance-regularity concepts.
Connected spectral properties with almost distance-regularity.
Provided examples from Foster census and posed open problems.
Abstract
Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called -walk-regularity. Another studied concept is that of -partial distance-regularity or, informally,…
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Advanced Algebra and Logic
