On symmetric association schemes and associated quotient-polynomial graphs
Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 947-969.

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d+1 distinct eigenvalues. Let 𝒜=𝒜(Γ) denote the subalgebra of Mat X () generated by A. We refer to 𝒜 as the adjacency algebra of Γ. In this paper we investigate algebraic and combinatorial structure of Γ for which the adjacency algebra 𝒜 is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) 𝒜 has a standard basis {I,F 1 ,...,F d }; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ with d+1 cells; (iii) the graph Γ is quotient-polynomial; and (iv) if we pick F{I,F 1 ,...,F d } then F has d+1 distinct eigenvalues if and only if span{I,F 1 ,...,F d }=span{I,F,...,F d }. We describe the combinatorial structure of quotient-polynomial graphs with diameter 2 and 4 distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether Γ is distance-regular or not and, more generally, which distance-i matrices are polynomial in A, giving also these polynomials.

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DOI: 10.5802/alco.187
Classification: 05E30,  05C50
Keywords: Symmetric association scheme, adjacency algebra, quotient-polynomial graph, intersection diagram.
Fiol, Miquel A. 1; Penjić, Safet 2

1 Departament de Matemàtiques Universitat Politécnica de Catalunya Barcelona Graduate School of Mathematics Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech) Catalonia, Spain
2 University of Primorska Andrej Marušič Institute Muzejski trg 2 6000 Koper, Slovenia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Fiol, Miquel A.; Penjić, Safet. On symmetric association schemes and associated quotient-polynomial graphs. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 947-969. doi : 10.5802/alco.187. https://alco.centre-mersenne.org/articles/10.5802/alco.187/

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