Abstract
In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.
Highlights
We consider undirected graphs without loops and multiple edges
For given vertex a of a graph Γ, we denote by Γi(a) the subgraph of Γ induced by the set of all vertices at distance i from a
Let Γ be a graph of diameter d and i ∈ {1, 2, 3, . . . , d}
Summary
We consider undirected graphs without loops and multiple edges. For given vertex a of a graph Γ, we denote by Γi(a) the subgraph of Γ induced by the set of all vertices at distance i from a. A graph Γ of diameter d is called a distance-regular graph with intersection array {b0, b1, . In the class of distance-regular graphs Γ of diameter 3, there are 5 hypothetical graphs with at most 28 vertices and non-integer eigenvalues. They have intersection arrays {18, 14, 5; 1, 2, 14}, {18, 15, 9; 1, 1, 10}, {21, 16, 10; 1, 2, 12}, {24, 21, 3; 1, 3.18}, and {27, 20, 7; 1, 4, 21}. We study the properties of a hypothetical distance-regular graph with intersection array {27, 20, 7; 1, 4, 21} and prove the following theorem. A distance-regular graph with intersection array {27, 20, 7; 1, 4, 21} does not exist
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