A graph vertex-induced subgraph ; sometimes called the ego graph in more recent literature) is isomorphic to (or to a member of) et al. 1989), so care is needed since the definition of local graphs excludes the vertex graph neighborhood and neighborhood graph include
For example, the only locally pentagonal (cycle graph 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 graphcubical graph cocktail party graph 16-cell graphcocktail party graph cocktail party graph complete graph complete graph complete complete 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 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 24-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 graphicosahedral graph 16-cell graphoctahedral graph 6-tetrahedral Johnson graph generalized quadrangle GQ(2,1)7-tetrahedral Johnson graph circulant graph 8-tetrahedral Johnson graph circulant graph 9-tetrahedral Johnson graph (3,6)-rook graph 10-tetrahedral Johnson graph (3,7)-rook graph tetrahedral graph triangle graph 5-triangular graph prism graph triangular graph rook 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 cyclotomic graph , 13-Paley graph cycle graph 24-Klein graph cubical graph 24-cell graph, circulant graph generalized quadrangle GQ(2,1)(4,4)-rook graph complement, 6-tetrahedral Johnson graph generalized quadrangle GQ(2,2)(8,2)-Kneser graph 5-halved cube graph Schläfli graph icosahedral graph 600-cell graph4-Möbius ladder 17-Paley graph 7-Möbius ladder 43-cyclotomic graph octahedral graph 16-cell graphPetersen graph Conway-Smith graph , Hall graph , (7,2)-Kneser graph prism graph 5-triangular graph prism graph 31-cyclotomic graph quartic vertex-transitive graph Qt31Schläfli graph Gosset graph square graph octahedral graph tetrahedral graph pentatope graph triangle graph tetrahedral graph triangular graph halved cube graph Hall-Janko graph utility graph complete tripartite graph
See also Graph Distance Graph ,
Graph Neighborhood ,
Local McLaughlin Graph ,
Locally Petersen Graph ,
M22 Graph ,
Suzuki Tower ,
Vertex-Induced Subgraph Explore with Wolfram|Alpha References Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. Referenced on Wolfram|Alpha Local Graph Cite this as: Weisstein, Eric W. "Local Graph." From MathWorld https://mathworld.wolfram.com/LocalGraph.html
Subject classifications More... Less... 
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 
(i.e. the vertex-induced subgraph; 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. 
) graph is the icosahedral graph (Brouwer et al. 1989, p. 5). 
-partite graph 
-partite graph 
graph
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