The Distance-regular Graphs with Intersection Number a1!=0 and with an Eigenvalue -1-(b1/2) (original) (raw)
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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.
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.
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].
European Journal of Combinatorics, 2008
In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ = −1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous), all its eigenvalues are integral, and the eigenvalue with multiplicity equal to the valency is either second largest or the smallest. Let x, y ∈ V Γ be two adjacent vertices, and z ∈ Γ 2 (x) ∩ Γ 2 (y). Then the intersection number τ 2 := |Γ (z) ∩ Γ 3 (x) ∩ Γ 3 (y)| is independent of the choice of vertices x, y and z. In the case of the coset graph of the doubly truncated binary Golay code, we have b 2 = τ 2 . We classify all the graphs with b 2 = τ 2 and establish that the just mentioned graph is the only example. In particular, we rule out an infinite family of otherwise feasible intersection arrays.
Distance–regular graphs having the M -property
Linear and Multilinear Algebra, 2012
We analyze when the Moore-Penrose inverse of the combinatorial Laplacian of a distanceregular graph is an 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 strongly regular graphs having the M -property and we give some families of distance regular graphs with diameter three that satisfy the M -property. Roughly speaking, we prove that all distance-regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance-regular graphs with diameter three, and no distance-regular graphs with diameter greater than three, have the M -property. In addition, we conjecture that no primitive distance-regular graph with diameter three has the M -property.
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.
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 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.