From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

Fiol, M. A.; Carriga, E

doi:10.1006/jctb.1997.1778

Cited by 83 publications

(127 citation statements)

References 22 publications

(42 reference statements)

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“…The following result can be seen as a version of the spectral excess theorem, due to Garriga and the author [13] (for short proofs, see Van Dam [5], and Fiol, Gago and Garriga [12]):…”

Section: Preliminariesmentioning

confidence: 98%

“…. , p d , introduced by the author and Garriga [13], are a sequence of orthogonal polynomials with respect to the inner product (1), normalized in such a way that p i 2 Γ = p i (k). Then, it is known that Γ is distance-regular if and only if such polynomials satisfy p i (A) = A i (the adjacency matrix of the distance-i graph Γ i ) for i = 0, .…”

Section: Preliminariesmentioning

confidence: 99%

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The spectral excess theorem for distance-regular graphs having distance- d graph with fewer distinct eigenvalues

Fiol

2015

Electronic Notes in Discrete Mathematics

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Let Γ be a distance-regular graph with diameter d and Kneser graph K = Γ d , the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distanceregular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.

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“…The following result can be seen as a version of the spectral excess theorem, due to Garriga and the author [13] (for short proofs, see Van Dam [5], and Fiol, Gago and Garriga [12]):…”

Section: Preliminariesmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

The spectral excess theorem for distance-regular graphs having distance- d graph with fewer distinct eigenvalues

Fiol

2015

Electronic Notes in Discrete Mathematics

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“…see again [7]. When the number a ( ) u only depends on , in which case we write a ( ) u = a ( ) , the graph G is called walk-regular (a concept introduced by Godsil and McKay in [12]).…”

Section: B Spectral Regularity and Walk-regularitymentioning

confidence: 98%

“…where {p h } 0≤h≤d are the so-called predistance polynomials, which are orthogonal with respect to the scalar product [7].…”

Section: The Geometry Of T-spreads In K-walk-regular Graphsmentioning

confidence: 99%

The geometry oft‐spreads ink‐walk‐regular graphs

Dalfó

Fiol

Garriga

2010

Journal of Graph Theory

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Abstract:A graph is walk-regular if the number of closed walks of length rooted at a given vertex is a constant through all the vertices for all . For a walk-regular graph G with d+1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its d-spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three-dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their t-spreads or vertices at distance t from each other. ᭧

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“…Thus, we have the following characterization which can be seen as the analogue of the spectral excess theorem [10,18,8] for edge-distance-regularity. …”

mentioning

confidence: 99%

Edge-distance-regular graphs

Cámara

Dalfó

Fàbrega

et al. 2011

Electronic Notes in Discrete Mathematics

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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.

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From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

Cited by 83 publications

References 22 publications

The spectral excess theorem for distance-regular graphs having distance- d graph with fewer distinct eigenvalues

The spectral excess theorem for distance-regular graphs having distance- d graph with fewer distinct eigenvalues

The geometry oft‐spreads ink‐walk‐regular graphs

Edge-distance-regular graphs

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