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Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Hadamard matrices and strongly regular graphs with the 3-e.c. adjacency property. | Hadamard matrices and strongly regular graphs with the 3-e.c. adjacency property Anthony Bonato Department of Mathematics Wilfrid Laurier University Waterloo Ontario Canada N2L 3C5 abonato@wlu.ca W. H. Holzmann Department of Mathematics and Computer Science University of Lethbridge Lethbridge Alberta Canada T1K 3M4 holzmann@uleth.ca Hadi Kharaghani Department of Mathematics and Computer Science University of Lethbridge Lethbridge Alberta Canada T1K 3M4 hadi@cs.uleth.ca Submitted July 2 2000 Accepted November 6 2000. Abstract A graph is 3-e.c. if for every 3-element subset S of the vertices and for every subset T of S there is a vertex not in S which is joined to every vertex in T and to no vertex in S T. Although almost all graphs are 3-e.c. the only known examples of strongly regular 3-e.c. graphs are Paley graphs with at least 29 vertices. We construct a new infinite family of 3-e.c. graphs based on certain Hadamard matrices that are strongly regular but not Paley graphs. Specifically we show The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada NSERC THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R1 1 that Bush-type Hadamard matrices of order 16n2 give rise to strongly regular 3-e.c. graphs for each odd n for which 4n is the order of a Hadamard matrix. Key words n-e.c. graphs strongly regular graphs adjacency property Bush-type Hadamard matrix design AMS subject classification Primary 05C50 Secondary 05B20. 1 Introduction Throughout all graphs are finite and simple. A strongly regular graph SRG v k A ỳ is a regular graph with v vertices of degree k such that every two joined vertices have exactly A common neighbours and every two distinct non-joined vertices have exactly ỳ common neighbours. For a fixed integer n 1 a graph G is n-existentially closed or n-e.c. if for every n-element subset S of the vertices and for every subset T of S there is a vertex not in S which is joined to every vertex in T and .
