On inertia and ratio type bounds for the k-independence number of a graph and their relationship
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...
| Authors: | , , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2023 |
| Country: | España |
| Institution: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repository: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:10459.1/463220 |
| Online Access: | https://doi.org/10.1016/j.dam.2023.03.015 https://hdl.handle.net/10459.1/463220 |
| Access Level: | Open access |
| Keyword: | Adjacency spectrum Independence number k-partially walk-regular k-power graph Mixed integer linear programming Polynomials |
| Summary: | 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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