---
tag: S6HMSC7N54
kind: concept
name: Graph of a polytope
aliases: [edge-graph-polytope, edge-graph-of-a-polytope, polytopal-graph, polytope-graph, skeleton-of-a-polytope, vertex-edge-graph-of-a-polytope]
sources:
  - wikipedia: en:74718534@r1367982734
    locator: Graph of a polytope
    note: adopted
    mention: wikipedia://en/74718534
    sections: [Summary]
same_as:
  wikidata: Q135553266
  wikipedia: Graph of a polytope
inventory: [9CERPXYJP8]
---

## Summary

In polytope theory, the **edge graph** (also known as **vertex-edge graph** or just **graph**) of a polytope is a combinatorial graph whose vertices and edges correspond directly to the vertices and edges of the polytope. As a purely [[combinatorial|0JKW7W7XFR]] object, the edge graph encodes incidence information, capturing which vertices are connected by edges, but it does not retain geometric data such as vertex positions or edge lengths. Further common names for the edge graph are **skeleton** and **1-skeleton**, though some authors reserve these terms for the geometric embedding formed by the vertices and edges in the polytope's ambient space. There is no universally agreed upon notation for the edge graph of a polytope $P$. Common notations include $G_P$, $G(P)$ or $\operatorname{skel}(P)$.