@MISC{Godsil11algebraicgraph, author = {Chris Godsil and Mike Newman}, title = {Algebraic Graph Theory}, year = {2011} }

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Abstract

Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix. A central topic and important source of tools is the theory of association schemes. An association scheme is, roughly speaking, a collection of graphs on a common vertex set which fit together in a highly regular fashion. These arise regularly in connection with extremal structures: such structures often have an unexpected degree of regularity and, because of this, often give rise to an association scheme. This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object. Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels. The theory of association schemes provides the most effective means for determining just how large is actually possible; this theory rests on Delsarte’s thesis [4], which showed how to use schemes to translate the problem into a question that be solved by linear programming.