Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The Strongly Regular (40, 12, 2, 4) Graphs. | The Strongly Regular 40 12 2 4 Graphs E. Spence Department of Mathematics University of Glasgow Glasgow G12 8QQ Scotland ted@ Submitted May 29 1998 Accepted April 20 2000 Abstract In a previous paper it was established that there are at least 27 non-isomorphic strongly regular 40 12 2 4 graphs. Using a different and more efficient method we have re-investigated these graphs and have now been able to determine them all and so complete the classification. We have discovered that there are precisely 28 non-isomorphic 40 12 2 4 strongly regular graphs. The one that was not found in the previous investigation is characterised uniquely by the fact that every neighbour graph is triangle-free. Key words and phrases Strongly regular graph classification AMS subject classifications Primary 05B05. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R22 2 1 Introduction A strongly regular 40 12 2 4 graph is a regular graph on 40 vertices of degree 12 such that each pair of adjacent vertices has 2 common neighbours and each pair of non-adjacent neighbours has 4 common neighbours. In 3 an incomplete enumeration of strongly regular 40 12 2 4 graphs established the existence of at least 27 all of which have at least one vertex x whose neighbour graph the subgraph induced by the vertices adjacent to x possesses a triangle. In the intervening years computers have speeded up considerably and this fact has aided the completion of the classification. Running the same program that was used in 3 the author has discovered that there are in fact 28 such strongly regular graphs so only one graph is missing from the original list. As a means of verifying the result a different search method was used. It is this that we describe briefly in the next section. The author is grateful to Brendan McKay for a further different and independent corroboration of the final result 2 . 2 The method As was first pointed out in 1 in any strongly regular 40 12 2 4 graph r the neighbour graph of
