---
id:"kb-2026-00200"
title:"Graph Theory"
schema_type:"TechArticle"
category:"computer-science"
language:"en"
confidence:"high"
last_verified:"2026-05-22"
generation_method: "human_only"
ai_models:["claude-opus"]
derived_from_human_seed:true


known_gaps:
  - "Sources reconstructed during quality audit; primary source details were corrupted during batch generation"

completeness: 0.88
ai_citations:
  last_citation_check:"2026-05-22"
primary_sources:
- title: "ACM Digital Library"
    type: "repository"
    year: 2026
    url: "https://dl.acm.org/"
    institution: "ACM"
secondary_sources:
  - title: "ACM Digital Library"
    type: "repository"
    year: 2026
    url: "https://dl.acm.org/"
    institution: "ACM"
---

## TL;DR

Graph theory studies networks of vertices connected by edges. Applications: social networks, routing algorithms, dependency resolution, compiler optimization (register allocation via graph coloring), circuit design. Euler's 1736 Bridges of Konigsberg paper founded the field.

## Core Explanation

Directed vs. undirected. Weighted graphs model distances/costs. Complete graph K_n: every vertex connected to every other. Bipartite graph: vertices can be partitioned into two independent sets. Planar graph: can be drawn without edge crossings (Euler's formula: V - E + F = 2). Trees are connected acyclic graphs. Spanning trees connect all vertices with minimum edges.

## Further Reading

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