Structure and uniqueness of the (81, 20, 1, 6) strongly regular graph

✍ Scribed by A.E. Brouwer; W.H. Haemers

• Book ID: 103060202
• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 409 KB
• Volume: 106-107
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90532-k

✦ Synopsis

We prove that there is a unique graph (on 81 vertices) with spectrum 20'2"'(-7)*".

We give several descriptions of this graph, and study its structure.

Let r = (X, E) be a strongly regular graph with parameters (v, k, ;1, p) = (81, 20, 1,. Then r (that is, its O-l adjacency matrix A) has spectrum 201260(-7)20, where the exponents denote multiplicities. We will show that up to isomorphism there is a unique such graph K More generally we give a short proof for the fact (due to Ivanov and Shpectorov ) that a strongly regular graph with parameters (v, k, A, p) = (q4, (q2 + l)(q -l), q -2, q(q -1)) that is the collinearity graph of a partial quadrangle (that is, in which all maximal cliques have size q) is the second subconstituent of the collinearity graph of a generalized quadrangle GQ(q, q*). In the special case q = 3 this will imply our previous claim, since A = 1 implies that all maximal cliques have size 3, and it is known (see Cameron et al. [5]) that there is a unique generalized quadrangle GQ(3,9) (and this generalized quadrangle has an automorphism group transitive on the points). The proof will use spectral techniques very much like those found in