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Moore graphs and beyond: A survey of the degree/diameter problem

Miller, M. and Siran, J. (2005). Moore graphs and beyond: A survey of the degree/diameter problem. Electronic Journal of Combinatorics, Dynamic Survey DS14

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Abstract

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds { called Moore bounds { for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attack- ing the degree/diameter problem `from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem `from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand. [brace not closed]

Item Type: Journal Article
ISSN: 1077-8926
Extra Information: Online journal, print version is known as Journal of Combinatorics (electronic ISSN 1077-8926, print ISSN 1097-1440)
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 8076
Depositing User: Jozef Širáň
Date Deposited: 14 Jun 2007
Last Modified: 02 Dec 2010 20:00
URI: http://oro.open.ac.uk/id/eprint/8076
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