On the multiplicity of eigenvalues of distance-regular graphs (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.
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.
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].
An Eigenvalue Characterization of Antipodal Distance-Regular Graphs
The Electronic Journal of Combinatorics
Let GGG be a regular (connected) graph with nnn vertices and d+1d+1d+1 distinct eigenvalues. As a main result, it is shown that GGG is an rrr-antipodal distance-regular graph if and only if the distance graph GdG_dGd is constituted by disjoint copies of the complete graph KrK_rKr, with rrr satisfying an expression in terms of nnn and the distinct eigenvalues.
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.
Linear and Multilinear Algebra
We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d + 1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.
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.