Abstract
Fiol has characterized quotient-polynomial graphs as precisely the connected graphs whose adjacency matrix generates the adjacency algebra of a symmetric association scheme. We show that a collection of non-negative integer parameters of size \(d + \frac{d(d-1)}{2}\) is adequate for describing symmetric association schemes of class d that are generated by the adjacency matrix of their first non-trivial relation. We use this to generate a database of the corresponding quotient-polynomial graphs that have small valency and up to 6 classes, and among these find new feasible parameter sets for symmetric association schemes with noncyclotomic eigenvalues.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data supporting this article is available at https://github.com/RoghayehMaleki/QPGdatabase-.
References
Arad, Z., Fisman, E., Muzychuk, M.: Generalized table algebras. Isr. J. Math. 114, 29–60 (1999)
Fiol, M.A.: Quotient-polynomial graphs. Linear Algebra Appl. 488, 363–376 (2016)
Fiol, M.A., Penjić, S.: On symmetric association schemes and associated quotient-polynomial graphs. Algebr. Comb. 4(6), 947–969 (2021)
Gavrilyuk, A.L., Vidali, J., Williford, J.S.: On few-class \(Q\)-polynomial association schemes: feasible parameters and nonexistence results. Ars Math. Contemp. 20, 103–127 (2021)
Hanaki, A.: Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes. Arch. Math. (Basel) 114(3), 259–263 (2020)
Hanaki, A.: Intersection numbers of integral standard generalized table algebras with non-cyclotomic minimal splitting fields (2016). http://math.shinshu-u.ac.jp/hanaki/pdf/teranishi.pdf
Herman, A., Maleki, R.: The search for small association schemes with noncyclotomic eigenvalues. Ars. Math. Contemp. 23, #P3.02 (2023)
Herman, A., Maleki, R.: QPGdatabase, August 31 (2023). https://github.com/RoghayehMaleki/QPGdatabase-
Ito, T.: Terwilliger algebras and the Weisfeiler–Lehman stabilization. In: Ivanov, A.A. (ed.) Algebraic Combinatorics and the Monster Group. LMS Lecture Note Series, vol. 487. Cambridge University Press, Cambridge (2024)
Penjić, S., Monzillo, G.: On commutative association schemes and associated (directed) graphs. arXiv:2307.11680v1 [math.CO]
Ponomarenko, I.: Some open problems for coherent configurations (2017). http://www.pdmi.ras.ru/inp/cp01.pdf
Xia, T.-T., Tan, Y.-Y., Liang, X., Koolen, J.H.: On association schemes generated by a relation or an idempotent. Linear Algebra Appl. 670, 1–18 (2023)
Funding
This study was funded by Natural Sciences and Engineering Research Council of Canada.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A. Herman’s work has been supported by NSERC. R. Maleki’s research is supported in part by the Ministry of Education, Science and Sport of Republic of Slovenia (University of Primorska Developmental funding pillar).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Herman, A., Maleki, R. Parameters of Quotient-Polynomial Graphs. Graphs and Combinatorics 40, 60 (2024). https://doi.org/10.1007/s00373-024-02789-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-024-02789-2