الصفوف 9–12Discrete Mathematicsمُخطَّط له

Graphs

Graphs as vertices and edges, degree and the handshake lemma, paths and connectivity, Euler circuits, trees and graph isomorphism.

math.discrete.graph-basics

لم يُكتب درس هذا الموضوع بعد. المخطط أدناه جزء من خريطة المنهج ولم تجرِ مراجعته.

يعتمد على

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 11

A simple graph consists of a set V of vertices and a set E of two-element subsets of V, called edges.
A tree is a connected graph with no cycles.
Graphs G and H are isomorphic if there is a bijection between their vertex sets that preserves adjacency in both directions.

Handshake lemma: the sum of the degrees of all vertices is 2|E|.
A connected graph has an Eulerian circuit if and only if every vertex has even degree.
A tree on n vertices has exactly n − 1 edges, and any two of its vertices are joined by a unique path.

Thinking a particular drawing determines the graph; very different drawings can show the same graph.
Confusing Euler circuits (using each edge once) with Hamiltonian cycles (visiting each vertex once).

Structure-preserving maps: homomorphisms and isomorphisms A graph isomorphism is a structure-preserving bijection whose inverse also preserves structure.
Quivers and path algebras Directed graphs (quivers) give rise to path algebras, whose modules are the representations of the quiver.