# generateGraphs -- generates the graphs on a given number of vertices

## Synopsis

• Usage:
G = generateGraphs n
G = generateGraphs(n, e)
G = generateGraphs(n, le, ue)
• Inputs:
• n, an integer, the number of vertices of the graphs, must be positive (see caveat)
• e, an integer, the number of edges in the graphs
• le, an integer, a lower bound on the number of edges in the graphs
• ue, an integer, an upper bound on the number of edges in the graphs
• Optional inputs:
• MaxDegree => an integer, default value null, an upper bound on the degrees of the vertices
• MinDegree => an integer, default value null, a lower bound on the degrees of the vertices
• Only4CycleFree => , default value false, whether to only allow graphs without 4-cycles
• OnlyBiconnected => , default value false, whether to only allow biconnected graphs
• OnlyBipartite => , default value false, whether to only allow bipartite graphs
• OnlyConnected => , default value false, whether to only allow connected graphs
• OnlyTriangleFree => , default value false, whether to only allow graphs without triangles (3-cycles)
• Outputs:
• G, a list, the graphs satisfying the input conditions

## Description

This method generates all graphs on n vertices subject to the constraints on the number of edges. It uses numerous options to allow further constraining of the output.

 ```i1 : generateGraphs(5, 4, 6, OnlyConnected => true) o1 = {D?{, DCw, DC{, DEw, DEk, DE{, DFw, DQo, DQw, DQ{, DUW, DUw, DTw} o1 : List```

## Caveat

The number of vertices n must be positive as nauty cannot handle graphs with zero vertices.