Summary: This appendix provides an index to all results regarding specific properties that were presented in this book. For each property, we provide references only to the sections (or statements) in which relevant results can be found.
The properties are partitioned into five main groups, whereas in each group the listing is by alphabetic order. The first list contains all properties of objects that are most naturally described as functions or sequences. The next three lists refer to the three models of testing graph properties, which were studied in Chapters 8–10. The last list refers to properties of distributions, which were studied in Chapter 11.
Properties of Functions. Such properties were studied mostly in Chapters 2–6.
affine functions: Last paragraph of Chapter 2.
affine subspaces: Section 126.96.36.199 as well as Exercises 5.9–5.11.
codewords: Chapter 13.
• For the Hadamard code see also linearity.
• For general linear codes see Proposition 1.11.
• For the Long Code see also dictatorship.
• For the Reed–Muller code see also low-degree polynomials.
dictatorship: Section 5.2.
homomorphism (a.k.a. group homomorphism): Chapter 2.
junta: Section 5.3 and Corollary 7.20.
linearity: Special case of homomorphism.
low-degree polynomials: Chapter 3.
majority: Proposition 1.1.
monomials: Section 5.2.2 as well as Corollaries 6.4 and 6.7.
monotonicity: Chapter 4. See also sorted.
proofs: Chapter 13.
sparse (low-degree) polynomials (and linear functions): Corollary 7.12.
sorted: Proposition 1.5 and 1.8. See also monotonicity.
Graph Properties (in the Dense Graph Model). All in Chapter 8.
biclique: Proposition 8.6.
bipartiteness: Section 8.3.1.
colorability (by fixed number of colors): Theorem 8.13.
degree regularity: Theorem 8.5.
induced subgraph freeness: Theorem 8.20.
max-clique: Special case of Theorem 8.12.
max-cut: Special case of Theorem 8.12.
min-bisection: Special case of Theorem 8.12.
subgraph freeness: Section 8.4.2.
Recall that properties of sparse graphs (e.g., planarity) and properties that are close to any graph (e.g., connectivity) are easy to test in this model.
Graph Properties (in the Bounded-Degree Graph Model). All in Chapter 9.
bipartiteness: Sections 9.3.1 and 9.4.1.
colorability (by three colors): Theorem 9.19.
connectivity: Section 9.2.3. See Section 9.2.4 for t-connectivity.
cycle-freeness: Section 9.2.5, Theorem 9.17, and Section 9.4.2.
degree regularity: Section 9.2.2.
Eulerian: See the last paragraph in Section 9.2.2 and the last paragraph in Section 9.2.3.
expansion: Theorem 9.18.
planarity: Special case of Theorem 9.25.
subgraph freeness: Theorem 9.4.