Distance–regular graphs having the M -property (original) (raw)
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M-Matrix Inverse problem for distance-regular graphs
We analyze when the Moore-Penrose inverse of the combinatorial Laplacian of a distance-regular graph is a M -matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore-Penrose inverse of the combinatorial Laplacian of a distance-regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance-regular graph has the M -property. We prove that only distance-regular graphs with diameter up to three can have the M -property and we give a characterization of the graphs that satisfy the M -property in terms of their intersection array. Moreover we exhaustively analyze the strongly regular graphs having the M -property and we give some families of distance regular graphs with diameter three that satisfy the M -property.
On almost distance-regular graphs
Journal of Combinatorial Theory, Series A, 2011
Distance-regular graphs have been a key concept in Algebraic Combinatorics and have given place 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 ( , 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 the existence of certain distance-regular graphs
Journal of Combinatorial Theory, Series B, 1982
Distance-regular graphs of valency > 2, diameter m, and girth 2m with the additional property that any two points having maximal distance belong to a unique 2m circuit are investigated. It is shown that such graphs can exist only if m < 3; if m = 3 only a finite number of valencies prove to be feasible.
On the distance spectrum of distance regular graphs
Linear Algebra and its Applications, 2015
The distance matrix of a simple graph G is D(G) = (d ij), where d ij is the distance between ith and jth vertices of G. The spectrum of the distance matrix is known as the distance spectrum or D-spectrum of G. A simple connected graph G is called distance regular if it is regular, and if for any two vertices x, y ∈ G at distance i, there are constant number of neighbors c i and b i of y at distance i − 1 and i + 1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d + 1 distinct D-eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D-eigenvalues. We also prove that distance regular graphs satisfying b i = c d−1 have at most d 2 + 2 distinct D-eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16].
Algebraic characterizations of distance-regular graphs
Discrete Mathematics, 2002
We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d + 1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied.
On distance-regular graphs with smallest eigenvalue at least -m
Journal of Combinatorial Theory, 2010
A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m⩾2m⩾2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersection number c2⩾2c2⩾2.
Partially Distance-regular Graphs and Partially Walk-regular Graphs
We study partially distance-regular graphs and partially walk- regular graphs as generalizations of distance-regular graphs and walk- regular graphs respectively. We conclude that the partially distance- regular graphs can be viewed as some extremal graphs of partially walk-regular graphs. In the special case that the graph is assumed to be regular with four distinct eigenvalues, a well known class of walk- regular graphs, we show that there exists a rational function f in the expression of the order and the four eigenvalues of the graph such that k2(x), the number of vertices with distance 2 to a vertex x, satisfles k2(x) ‚ f; furthermore we show the equality holds for each vertex x if and only if the graph is distance-regular with diameter 3: Keywords: Partially distance-regular graphs; partially walk-regular
Journal of Combinatorial Theory, Series A, 2011
Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edgedistance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.
2014
This is a survey of distance-regular graphs. We present an introduction to distanceregular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN'
Journal of Algebraic Combinatorics, 2001
We consider a distance-regular graph Γ with diameter d ≥ 3 and eigenvalues k = θ0 > θ1 > ... > θd . We show the intersection numbers a 1, b 1 satisfy \left( {\theta _1 + \frac{k}{{a_1 + 1}}} \right)\left( {\theta _d + \frac{k}{{a_1 + 1}}} \right) \geqslant - \frac{{ka_1 b_1 }}{{(a_1 + 1)^2 }}.$$ We say Γ is tight whenever Γ is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show Γ is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show Γ is tight if and only if a 1 ≠ 0, a d = 0, and Γ is 1-homogeneous in the sense of Nomura. We show Γ is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues −1 − b 1(1 + θ1)−1 and −1 − b 1(1 + θd )−1. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.