On almost distance-regular graphs

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 $m$-walk-regularity. Another studied concept is that of $m$-partial distance-regularity or, informally, distance-regularity up to distance $m$. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of $(\ell,m)$-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem

On Almost Distance-Regular Graphs

2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial

On Almost Distance-Regular Graphs

Preprin

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

Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first propose two dual concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as partially distance-regular graphs. Our study focuses on finding out when almost distance-regularity leads to distance-regularity. In particular, some ‘economic’ (in the sense of minimizing the number of conditions) old and new characterizations of distance-regularity are discussed. Moreover, other characterizations based on the average intersection numbers and the recurrence coefficients are obtained. In some cases, our results can also be seen as a generalization of the so-called spectral excess theorem for distance-regular graphs.Peer Reviewe

When almost distance-regularity attains distance-regularity

Generally speaking, `almost distance-regular graphs' are graphs which share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we rst propose four basic di erent (but closely related) concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as walk-regular graphs and partially distance-regular graphs. Here it is always assumed that the diameter D of the graph attains its maximum possible value allowed by its number d+1 of di erent eigenvalues; that is, D = d, as happens in every distance-regular graph. Our study focuses on nding out when almost distance- regularity leads to distance-regularity. In other words, some `economic' (in the sense of minimizing the number of conditions) old and new characterizations of distance- regularity are discussed. For instance, if A0;A1; : : : ;AD and E0;E1; : : : ;Ed denote, respectively, the distance matrices and the idempotents of the graph; and D and A stand for their respective linear spans, any of the two following `dual' conditions su ce: (a) A0;A1;AD 2 A; (b) E0;E1;Ed 2 D. Moreover, other characterizations based on the preintersection parameters, the average intersection numbers and the recurrence coe cients are obtained. In some cases, our results can be also seen as a generalization of the so-called spectral excess theorem for distance-regular graphs.Postprint (published version

Locally Recoverable Codes From Planar Graphs

In this paper we apply Kadhe and Calderbank's definition of LRCs from convex polyhedra and planar graphs [4] to analyze the codes resulting from 3-connected regular and almost regular planar graphs. The resulting edge codes are locally recoverable with availability two. We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes. Constructions of regular and almost regular planar graphs are given, and their associated code parameters are determined. In certain cases, the code families meet the rate bound