Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994 Association of Mizar Users

## The Formalization of Simple Graphs

Yozo Toda
Information Processing Center, Chiba University

### Summary.

A graph is simple when \begin{itemize} \parskip -1mm \item it is non-directed, \item there is at most one edge between two vertices, \item there is no loop of length one. \end{itemize} A formalization of simple graphs is given from scratch. There is already an article [10], dealing with the similar subject. It is not used as a starting-point, because [10] formalizes directed non-empty graphs. Given a set of vertices, edge is defined as an (unordered) pair of different two vertices and graph as a pair of a set of vertices and a set of edges.\par The following concepts are introduced: \begin{itemize} \parskip -1mm \item simple graph structure, \item the set of all simple graphs, \item equality relation on graphs. \item the notion of degrees of vertices; the number of edges connected to, or the number of adjacent vertices, \item the notion of subgraphs, \item path, cycle, \item complete and bipartite complete graphs, \end{itemize}\par Theorems proved in this articles include: \begin{itemize} \parskip -1mm \item the set of simple graphs satisfies a certain minimality condition, \item equivalence between two notions of degrees. \end{itemize}

#### MML Identifier: SGRAPH1

The terminology and notation used in this paper have been introduced in the following articles [12] [7] [15] [13] [2] [1] [4] [5] [6] [3] [9] [8] [14] [11]

#### Contents (PDF format)

1. Preliminaries
2. Simple Graphs
3. Equality Relation on Simple Graphs
4. Properties of Simple Graphs
5. Subgraphs
6. Degree of Vertices
7. Path and Cycle
8. Some Famous Graphs

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