Abstract
A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array$$\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\} (r, t \geq 1),$$$\{135,\! 128,\! 16; 1,\! 16,\! 120\}$, $\{234,\! 165,\! 12; 1,\! 30,\! 198\}$ or $\{55,\! 54,\! 50,\! 35,\! 10; 1,\! 5,\! 20,\! 45,\! 55\}$. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence.
Highlights
Distance-regular graphs were introduced around 1970 by N
A number of feasibility conditions for distance-regular graphs have been found, which allows us to compile a list of feasible intersection arrays for small distance-regular graphs, see Brouwer et al [2, 3, 4] and Williford [22]
In this paper we give proofs of nonexistence for distance-regular graphs belonging to a two-parameter infinite family, as well as for graphs with intersection arrays
Summary
Distance-regular graphs were introduced around 1970 by N. The electronic journal of combinatorics 25(4) (2018), #P4.21 existence, so proofs of nonexistence of distance-regular graphs with feasible intersection arrays are a contribution to the classification. In this paper we give proofs of nonexistence for distance-regular graphs belonging to a two-parameter infinite family, as well as for graphs with intersection arrays. Sage is free open-source software written in the Python programming language [17], with many functionalities deriving from other free open-source software, such as Maxima [16], which Sage uses for symbolic computation. The sage-drg package is free opensource software available under the MIT license, written in the Python programming language, making use of the Sage library. The sage-drg package can be seen as a move from closed-source proprietary software to free open-source software, which allows one to check all code for correctness, making the results verifiable
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