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

Aida Abiad; Cristina Dalfó; Miquel Àngel Fiol; Sjanne Zeijlemaker

doi:10.1016/j.dam.2023.03.015

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

Aida Abiad, Cristina Dalfó (Corresponding author), Miquel Àngel Fiol, Sjanne Zeijlemaker

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Abstract

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(=α₁) 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.

Original language	English
Pages (from-to)	96-109
Number of pages	14
Journal	Discrete Applied Mathematics
Volume	333
DOIs	https://doi.org/10.1016/j.dam.2023.03.015
Publication status	Published - 15 Jul 2023

Bibliographical note

Funding Information:
The research of Aida Abiad is partially supported by the FWO, Belgium grant 1285921N . The research of Cristina Dalfó and Miquel Àngel Fiol has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00 .

Funding

The research of Aida Abiad is partially supported by the FWO, Belgium grant 1285921N . The research of Cristina Dalfó and Miquel Àngel Fiol has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00 .

Keywords

Adjacency spectrum
Independence number
k-partially walk-regular
k-power graph
Mixed integer linear programming
Polynomials

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Cite this

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title = "On inertia and ratio type bounds for the k-independence number of a graph and their relationship",

abstract = "For k≥1, the k-independence number αk 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(=α1) of a graph, due to Cvetkovi{\'c} 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 αk, 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.",

keywords = "Adjacency spectrum, Independence number, k-partially walk-regular, k-power graph, Mixed integer linear programming, Polynomials",

author = "Aida Abiad and Cristina Dalf{\'o} and Fiol, \{Miquel {\`A}ngel\} and Sjanne Zeijlemaker",

note = "Funding Information: The research of Aida Abiad is partially supported by the FWO, Belgium grant 1285921N . The research of Cristina Dalf{\'o} and Miquel {\`A}ngel Fiol has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00 . ",

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AU - Fiol, Miquel Àngel

AU - Zeijlemaker, Sjanne

N1 - Funding Information: The research of Aida Abiad is partially supported by the FWO, Belgium grant 1285921N . The research of Cristina Dalfó and Miquel Àngel Fiol has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00 .

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N2 - For k≥1, the k-independence number αk 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(=α1) 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 αk, 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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KW - Independence number

KW - k-partially walk-regular

KW - k-power graph

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KW - Polynomials

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