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The first chapter contains the basics of the theory of strongly regular graphs. In particular all basic notions such as parameters and spectrum are rigorously defined. The standard example such as Johnson graphs, Hamming graphs, Paley graphs are introduced. We treat Seidel switching and regular two-graphs, (induced) subgraphs, strongly regular graphs with smallest eigenvalue —2, regular partitions and regular (intriguing) sets. We enumerate the small examples and discuss prolific constructions. This chapter also contains an introduction to two slightly more general objects needed in the book: distance regular graphs (including the main examples, and a discussion on imprimitivity), and association schemes and coherent configurations (including a brief discussion of the Bose-Mesner algebra, linear programming bound, code-clique theorem, Krein parameters, Euclidean representation, subschemas, the absolute bound and the mu-bound).

We prove that the class of reflexive asymptotic-$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_{0}$ space $Y$, then $X$ is also reflexive and asymptotic-$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

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A NEW COARSELY RIGID CLASS OF BANACH SPACES

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2 results

1 - Graphs

Summary

A NEW COARSELY RIGID CLASS OF BANACH SPACES