tailieunhanh - Báo cáo toán học: "A note on constructing large Cayley graphs of given degree and diameter by voltage assignments"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A note on constructing large Cayley graphs of given degree and diameter by voltage assignments. | A note on constructing large Cayley graphs of given degree and diameter by voltage assignments Ljiljana Brankovic Mirka Miller Department of Computer Science and Software Engineering The University of Newcastle NSW 2308 Australia e-mail lbrankov mirka @ Jan Plesnik Department of Numerical and Optimization Methods Faculty of Mathematics and Physics Comenius University 842 15 Bratislava Slovakia e-mail plesnik@ Joe Ryan Department of Mathematics The University of Newcastle NSW 2308 Australia e-mail joe@ Jozef Sirán Department of Mathematics SvF Slovak Technical University Radlinskeho 11 813 68 Bratislava Slovakia e-mail siran@ Submitted July 7 1997 Accepted August 8 1997. Abstract Voltage graphs are a powerful tool for constructing large graphs called lifts with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree diameter problem which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree 15 and diameter 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups with voltages in other cyclic groups. This research started when J. Plesnik and J. Sirán were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995 supported by small ARC grant. 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R9 2 This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter. AMS Subject Classification 05C25. 1 Introduction The problem of finding for given d and k the largest order nd k of a graph of maximum degree d and diameter k is well known as the degree diameter problem. An obvious upper bound on nd k is the Moore bound Md k