The geometry of<i>t</i>‐spreads in<i>k</i>‐walk‐regular graphs

Dalfó, C.; Fiol, M. A.; Garriga, E.

doi:10.1002/jgt.20458

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Maximally Independent Sets1

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2022

2023

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Cited by 2 publications

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References 14 publications

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“…Next, we extend a result by Dalfó, Fiol, and Garriga [16] by showing that every antipodal distance-regular graph with odd diameter is a tight (d − 1)-CH graph (see CH2).…”

Section: The Values Xmentioning

confidence: 52%

See 1 more Smart Citation

On inertia and ratio type bounds for the $k$-independence number of a graph and their relationship

Abiad¹,

Dalfó²,

Fiol³

et al. 2022

Preprint

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For k ≥ 1, the k-independence number α k of a graph is the maximum number of vertices that are mutually at distance greater than k. The well-known inertia and ratio bounds for the (1-)independence number α(= α 1 ) of a graph, due to Cvetković and Hoffman, respectively, were generalized recently for every value of k. We show that, for graphs with enough regularity, the polynomials involved in such generalizations are closely related and give exact values for α k , showing a new relationship between the inertia and ratio type bounds. Additionally, we investigate the existence and properties of the extremal case of sets of vertices that are mutually at maximum distance for walk-regular graphs. Finally, we obtain new sharp inertia and ratio type bounds for partially walk-regular graphs by using the predistance polynomials.

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“…Next, we extend a result by Dalfó, Fiol, and Garriga [16] by showing that every antipodal distance-regular graph with odd diameter is a tight (d − 1)-CH graph (see CH2).…”

Section: The Values Xmentioning

confidence: 52%

“…Thus, the (d − 1)-independence number α d−1 coincides with the so-called the dclique number or d-spread number ω d . In this context, Dalfó, Fiol, and Garriga [16] proved the following result.…”

Section: Maximally Independent Setsmentioning

confidence: 86%