Chapter 11
Basics of Graph Theory
Summary.
graphs are defined by sets, not by diagrams
deleting a vertex or edge
how to use proofs by induction on graphs
Euler's handshaking lemma
graph invariants, distinguishing between graphs
labeled and unlabeled graphs
random graphs
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Important definitions:
graph, vertex, edge
loop, multiple edge, multigraph, simple graph
endvertex, incident, adjacent, neighbour
degree, valency, isolated vertex
subgraph, induced subgraph
complete graph, complement of a graph
isomorphic graphs, isomorphism between graphs
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Notation:
\(\displaystyle u \sim v\)
\(\displaystyle uv\)
val\((v)\text{,}\) deg\((v)\text{,}\) \(d(v)\text{,}\) \(d_G(v)\)
\(G\setminus\{v\}\text{,}\) \(G\setminus\{e\}\)
\(\displaystyle K_n, G^c\)
\(\displaystyle G_1 \cong G_2\)