The GraphTheory[SpecialGraphs] subpackage collects routines for generating well-known graphs that are named in the literature, such as the PetersenGraph.

The commands for generating complete graphs, paths, and cycles on  vertices are not in the SpecialGraphs subpackage: they are part of the main GraphTheory package.  Follow the hyperlinks in the previous sentence to reach the help pages for the respective commands.

The following is a list of special graphs that can be generated.

AntiPrismGraph

CageGraph

ClebschGraph

CompleteBinaryTree

CompleteKaryTree

CoxeterGraph

DesarguesGraph

DodecahedronGraph

DoubleStarSnark

DyckGraph

FlowerSnark

FosterGraph

GeneralizedBlanusaSnark

GeneralizedHexagonGraph

GeneralizedPetersenGraph

GoldbergSnark

GridGraph

GrinbergGraph

GrotzschGraph

HeawoodGraph

HerschelGraph

HoffmanSingletonGraph

HypercubeGraph

IcosahedronGraph

KneserGraph

LCFGraph

LeviGraph

McGeeGraph

MobiusKantorGraph

OctahedronGraph

OddGraph

PappusGraph

PayleyGraph

PetersenGraph

PrismGraph

RobertsonGraph

ShrikhandeGraph

SoccerBallGraph

StarGraph

SzekeresSnark

TetrahedronGraph

ThetaGraph

TorusGridGraph

Tutte8CageGraph

WebGraph

WheelGraph