Restrictions on classical distance-regular graphs

Jurišić, Aleksandar; Vidali, Janoš

doi:10.1007/s10801-017-0765-3

Cited by 7 publications

(8 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Distance-regular graphs with classical parameters The eigenvalues of distance-regular graphs with classical parameters (d, q, α, β) are given in Jurišić and Vidali [11] (see Lemma 2). In particular, θ 0 = [d] q β and θ 1 = [d − 1] q (β − α) − 1, which implies…”

Section: Grassmann Graphsmentioning

confidence: 99%

Of Shadows and Gaps in Spatial Search

Chan¹,

Godsil²,

Tamon³

et al. 2022

Preprint

View full text Add to dashboard Cite

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty et al. (Physical Review A, 102:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

show abstract

Section: Grassmann Graphsmentioning

confidence: 99%

Of Shadows and Gaps in Spatial Search

Chan¹,

Godsil²,

Tamon³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We call these numbers triple intersection numbers. They have first been studied in the case of strongly regular graphs [15], and later also for distance-regular graphs, see for example [18,36,37,38,58]. Unlike the intersection numbers, these numbers may depend on the particular choice of vertices u, v, w and not only on their pairwise distances.…”

Section: Preliminariesmentioning

confidence: 99%

Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

Vidali

2018

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array ,128,16; 1,16,120}, {234,165,12; 1,30,198} or {55,54,50,35,10; 1,5,20,45,55}. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence. {135

show abstract

“…On the other hand, the Q-polynomial property allows us to consider triple intersection numbers with respect to some triples of vertices, which can be thought of as a generalization of intersection numbers to triples of starting vertices instead of pairs. This technique has been previously used by various researchers [8,10,17,21,22,23,36,37], mostly to prove nonexistence of some strongly regular and distance-regular graphs with equality in the so-called Krein conditions, in which case combining the restrictions implied by the triangle inequality with triple intersection numbers seems the most fruitful. Yet, while calculating triple intersection numbers when the P -polynomial property is absent is harder, we managed to rule out a number of open cases from the tables.…”

Section: Introductionmentioning

confidence: 99%