Local Graph
A graph 
is said to be locally 
, where 
is a graph (or class of graphs), when for every vertex 
, the graph induced on 
by the set of adjacent vertices of 
(sometimes called the ego graph in more recent literature) is isomorphic to (or to a member of) 
. Note that the term "neighbors" is sometimes used instead of "adjacent vertices" here (e.g., Brouwer et al. 1989), so care is needed since the definition of local graphs excludes the vertex 
on which a subgraph is induced, while the definitions of graph neighborhood and neighborhood graph include 
itself.
For example, the only locally pentagonal (cycle graph 
) graph is the icosahedral graph (Brouwer et al. 1989, p. 5).
The following table summarizes some named graphs that have named local graphs.
| graph |
local graph |
| 24-cell graph |
cubical graph |
cocktail party graph  |
16-cell graph |
cocktail party graph  |
cocktail party graph  |
complete graph  |
complete graph  |
complete  -partite graph  |
complete  -partite graph  |
| Conway-Smith graph |
Petersen graph |
| 19-cyclotomic graph |
cycle graph  |
| 31-cyclotomic graph |
prism graph  |
| 37-cyclotomic graph |
 |
| 43-cyclotomic graph |
7-Möbius ladder graph |
| 64-cyclotomic graph |
(3,7)-rook graph |
generalized hexagon  |
circulant graph  |
generalized octagon  |
 |
| Gosset graph |
Schläfli graph |
| Hall graph |
Petersen graph |
| Hall-Janko graph |
 graph |
 -halved cube graph |
 -triangular graph |
 -Hamming graph |
circulant graph  |
| line graph of the Hoffman-Singleton graph |
circulant graph  |
| icosahedral graph |
cycle graph  |
| (8,4)-Johnson graph |
(4,4)-rook graph |
| (9,4)-Johnson graph |
(4,5)-rook graph |
| Klein graph |
cycle graph  |
| (7,2)-Kneser graph |
Petersen graph |
| (8,2)-Kneser graph |
generalized quadrangle GQ(2,2) |
 -Kneser graph |
 -Kneser graph |
| (10,3)-Kneser graph |
odd graph  |
 -rook graph |
circulant graph  |
| (4,4)-rook graph complement |
generalized quadrangle GQ(2,1) |
| octahedral graph |
square graph |
| 13-Paley graph |
cycle graph  |
| 17-Paley graph |
4-Möbius ladder |
| 25-Paley graph |
circulant graph  |
| 29-Paley graph |
circulant graph  |
| pentatope graph |
tetrahedral graph |
| Schläfli graph |
5-halved cube graph |
| Shrikhande graph |
cycle graph  |
| 600-cell graph |
icosahedral graph |
| 16-cell graph |
octahedral graph |
| 6-tetrahedral graph |
generalized quadrangle GQ(2,1) |
| 7-tetrahedral graph |
circulant graph  |
| 8-tetrahedral graph |
circulant graph  |
| 9-tetrahedral graph |
(3,6)-rook graph |
| 10-tetrahedral graph |
(3,7)-rook graph |
| tetrahedral graph |
triangle graph |
| 5-triangular graph |
prism graph  |
 -triangular graph |
 -rook graph |
 graph |
quartic vertex-transitive graph Qt31 |
The following table gives a list of some local graphs and graphs in which they are contained.
local graph  |
graphs containing  |
 |
37-cyclotomic graph |
 |
generalized hexagon GH(3,1), generalized octagon GO(3,1), (4,4)-rook graph |
 |
(3,4)-Hamming graph |
 |
(4,4)-Hamming graph |
cycle graph  |
icosahedral graph |
cycle graph  |
Shrikhande graph, circulant graph  ,  ,  ,  , 19-cyclotomic graph, 13-Paley graph |
cycle graph  |
Klein graph |
| cubical graph |
24-cell graph, circulant graph  |
| generalized quadrangle GQ(2,1) |
(4,4)-rook graph complement, 6-tetrahedral graph |
| generalized quadrangle GQ(2,2) |
(8,2)-Kneser graph |
| 5-halved cube graph |
Schläfli graph |
| icosahedral graph |
600-cell graph |
| 4-Möbius ladder |
17-Paley graph |
| 7-Möbius ladder |
43-cyclotomic graph |
| octahedral graph |
16-cell graph |
| Petersen graph |
Conway-Smith graph, Hall graph, (7,2)-Kneser graph |
prism graph  |
5-triangular graph |
prism graph  |
31-cyclotomic graph |
| quartic vertex-transitive graph Qt31 |
 graph |
| Schläfli graph |
Gosset graph |
square graph  |
octahedral graph |
| tetrahedral graph |
pentatope graph |
triangle graph  |
tetrahedral graph |
 -triangular graph |
 -halved cube graph |
 graph |
Hall-Janko graph |
| utility graph |
complete tripartite graph  |
REFERENCES
Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, pp. 4-5, 256, and 434, 1989.
