Graph Theory: Basic Definitions
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# Graph Theory: Basic Definitions ## 5.1 Basic Definitions A **graph** $G = (V, E)$ consists of: - $V$ = set of vertices (nodes) - $E$ = set of edges (pairs of vertices) **Types:** - **Simple graph:** No loops or multiple edges - **Directed graph (digraph):** Edges have direction - **Weighted graph:** Edges have wei...
Graph Theory: Basic Definitions
5.1 Basic Definitions
A graph consists of:
- = set of vertices (nodes)
- = set of edges (pairs of vertices) Types:
- Simple graph: No loops or multiple edges
- Directed graph (digraph): Edges have direction
- Weighted graph: Edges have weights
5.2 Degree and Handshaking Lemma
Degree = number of edges incident to .
Corollary: Every graph has an even number of odd-degree vertices.
5.3 Paths and Cycles
- Walk: Sequence of vertices where consecutive vertices are adjacent
- Path: Walk with no repeated vertices
- Cycle: Path that starts and ends at same vertex
- Connected: For every pair of vertices, there's a path between them
- Component: Maximal connected subgraph
5.4 Special Graphs
| Graph | Notation | Properties |
|---|---|---|
| Complete graph | edges, every pair adjacent | |
| Cycle | vertices forming a single cycle | |
| Path | vertices in a line | |
| Bipartite | Two parts, all edges between parts | |
| Complete bipartite | All possible edges between parts |