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

Brouwer, A.E, W.H. Haemers, Structure and uniqueness of the (81,20, 1.6) strongly regular graph, Discrete Mathematics 106/107 (1992) 77-82. 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, 6). 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 [9]) 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 Correspondence to: A.E. Brouwer, Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, Netherlands. 0012-365X/92/$05.00