A Computer-Assisted Proof of the Uniqueness of the Perkel Graph

Abstract

The Perkel graph is a distance-regular graph of order 57, degree 6 and diameter 3, with intersection array (6, 5, 2; 1, 1, 3). We describe a computer assisted proof that every graph with this intersection array is isomorphic to the Perkel graph. The computer proof relies heavily on the fact that the minimal idempotents for , and their submatrices, are positive semidefinite. To minimize the risk of computer errors we have used two different methods to establish the same theorem and as an added precaution large parts of the corresponding programs were written by different authors. The first method generates plausible subgraphs induced by all vertices at distance 3 from a fixed vertex of and then tries to extend each of the generated graphs to a full graph with the given intersection array. The second method generates possible neighborhoods for a pentagon in . It turns out that every such pentagon can be extended to a Petersen graph in . We then prove mathematically that there is, up to isomorphism, only a single graph with this property.

Characterizing distance-regularity of graphs by the spectrum

“…In [7] it was (among other things) proved that the M 22 graph is the unique graph with spectrum { [7] 1 , [4] 55 , [1] 154 , [−3] 99 , [−4] 21 } by showing that a graph with that spectrum must be distanceregular. Here we will do the same for the Ivanov-Ivanov-Faradjev graph: we will show that a graph with spectrum [7] 1 , [5] 42 , [4] must be distance-regular, and hence is the Ivanov-Ivanov-Faradjev graph. The proof of this result uses the following two lemmas.…”

An Alternative Way to Generalize the Pentagon

“…Since the number of points is , we have , and so we see that the intersection array of G is . Lemma There is no pentagonal geometry of order (6, 10) with distance‐regular diameter 3 deficiency graph. Proof The deficiency graph Γ would have parameters and by a result of Coolsaet and Degraer , the Perkel graph on 57 vertices is the unique distance‐regular graph with these parameters. Note that points on a nonopposite line are at distance at least 3 in G , by Lemma 2.7, so nonopposite lines form cliques of size 3 in the “distance at least 3” graph obtained from G .…”

On 3-chromatic distance-regular graphs

“…See [2], p. 402 for the structure (and 3-coloring), and [5] for uniqueness. (A slightly nicer description: take vertex set Z 3 × Z 19 and join (i, j) to (i + 1, k) when (k − j) 3 = 2 6i .)…”

“…The resulting graph is a near hexagon, uniquely determined by its intersection array (see [1]). Case k) is the case where C is the dual of the perfect ternary Golay code (an [11,5,6] 3 code). We show below in Theorem 3.9 that also this graph is uniquely determined by its parameters.…”