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Graph Theory Textbook
From the definition of a graph to planarity, coloring, and matroids. A textbook series building definitions, theorems, and proofs in a systematic progression.
Linear Algebra Textbook
From the axioms of vector spaces to Jordan normal form. A textbook series building definitions, theorems, and proofs in a systematic progression.
Group Theory Textbook
From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
Number Theory Textbook
From divisibility and congruences to p-adic numbers and algebraic integers. A textbook series building definitions, theorems, and proofs in a systematic progression.
Popular Articles
Cosets and quotient groups: the art of controlled forgetting
The quotient group G/N is the first major conceptual hurdle in group theory. By computing small examples by hand -- from clock arithmetic to D4 modulo its center -- we build the intuition for what "dividing" a group really means, and why normality is indispensable.
Paths and Connectivity
We formalize the notions of walks, paths, and cycles, then develop the theory of connected components, vertex and edge connectivity, cut vertices and bridges, the characterization of bipartite graphs, and Menger's theorem.
Cosets and Lagrange's Theorem
Beginning with the definition of cosets of a subgroup, we prove Lagrange's theorem --- the assertion that the order of a subgroup divides the order of the group. As applications, we derive Fermat's little theorem and Euler's theorem, and we exhibit the alternating group A_4 as a counterexample to the converse.
Directed Graphs and Topological Sorting
Strongly connected components (SCCs), Tarjan's and Kosaraju's algorithms, DAGs and topological sorting, the correspondence with partial orders, and dynamic programming on DAGs.
Trees and Forests: The Minimal Connected Structures
We prove the equivalent characterizations of trees, establish the formula $|E|=|V|-1$, introduce rooted trees and Cayley's formula, and develop the theory of spanning trees and minimum spanning trees via Kruskal's and Prim's algorithms.
Fundamental Definitions and Basic Properties of Graphs
Starting from the formal definitions of undirected and directed graphs, we develop the notions of degree, the handshaking lemma, complete graphs $K_n$, complete bipartite graphs $K_{m,n}$, and graph isomorphism, with full proofs throughout.
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