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and the 24 other vectors obtained by permuting the first six coordinates of these three vectors. These 27 vectors correspond to the vertices of the Schläfli graph; two vertices are adjacent if and only if the corresponding two vectors have 1 as their
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of any vertex in the Schläfli graph forms a 16-vertex subgraph in which each vertex has 10 neighbors (the numbers 16 and 10 coming from the parameters of the Schläfli graph as a strongly regular graph). These subgraphs are all
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subgraphs of the product. The Schläfli graph has a total of 36 subgraphs of this form, one of which consists of the zero-one vectors in the eight-dimensional representation described above.
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is countably ultrahomogeneous. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement. The proof relies on the
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474:. Note that Cameron and van Lint use an alternative definition of these graphs in which both the Schläfli graph and the Clebsch graph are
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of the Schläfli graph. That is, two vertices are adjacent in the Schläfli graph if and only if the corresponding pair of lines are
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Bussemaker, F. C.; Neumaier, A. (1992), "Exceptional graphs with smallest eigenvalue-2 and related problems",
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in which the two lines have disjoint neighborhoods. Correspondingly, in the Schläfli graph, each edge
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187:. This symmetric projection contains 2 rings of 12 vertices, and 3 vertices coinciding at the center.
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588:, London Mathematical Society student texts, vol. 22, Cambridge University Press, p. 35,
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The Schläfli graph may also be constructed from the system of eight-dimensional vectors
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of the whole graph. If a graph is 5-ultrahomogeneous, it is ultrahomogeneous for every
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Alternately, this graph can be seen as the complement of the collinearity graph of the
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273:. It plays an important role in the structure theory for claw-free graphs by
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Classification of some homogeneous and ultrahomogeneous structures
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Cameron, Peter
Jephson; van Lint, Jacobus Hendricus (1991),
224:(−1/2, −1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2),
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Cameron, Peter
Jephson (1980), "6-transitive graphs",
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with parameters srg(27, 16, 10, 8).
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287:, a set of 12 lines whose intersection graph is a
280:Any two skew lines of these 27 belong to a unique
179:The Schläfli graph is seen as a 1-skeleton of the
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295:belongs uniquely to a subgraph in the form of a
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636:, Ph.D. thesis, Université Libre de Bruxelles
610:(2005), "The structure of claw-free graphs",
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559:Journal of Combinatorial Theory, Series B
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163:with 27 vertices and 216 edges. It is a
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586:Designs, graphs, codes and their links
516:, D.Phil. thesis, University of Oxford
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410:classification of finite simple groups
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642:Holton, D. A.; Sheehan, J. (1993),
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549:10.1090/S0025-5718-1992-1134718-6
460:Bussemaker & Neumaier (1992)
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376:vertices can be extended to an
275:Chudnovsky & Seymour (2005)
133:Table of graphs and parameters
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613:Surveys in combinatorics 2005
472:Cameron & van Lint (1991)
265:. Since the Clebsch graph is
221:(1, 0, 0, 0, 0, 0, 0, 1), and
572:10.1016/0095-8956(80)90063-5
478:from their definitions here.
445:Holton & Sheehan (1993)
244:Subgraphs and neighborhoods
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650:Cambridge University Press
528:Mathematics of Computation
632:Devillers, Alice (2002),
512:Buczak, J. M. J. (1980),
396:, 3 × 3
357:A graph is defined to be
218:(1, 0, 0, 0, 0, 0, 1, 0),
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684:Andries E. Brouwer page.
388:graphs of this type are
321:{\displaystyle \square }
269:, the Schläfli graph is
709:Strongly regular graphs
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238:generalized quadrangle
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165:strongly regular graph
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195:of the 27 lines on a
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342:belong to different
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201:locally linear graph
540:1992MaCom..59..583B
514:Finite Group Theory
368:between two of its
334:in such a way that
282:Schläfli double six
667:Weisstein, Eric W.
652:, pp. 270–271
645:The Petersen Graph
384:; the only finite
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193:intersection graph
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699:Individual graphs
604:Chudnovsky, Maria
595:978-0-521-41325-1
370:induced subgraphs
362:-ultrahomogeneous
297:Cartesian product
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16:(Redirected from
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670:"Schläfli Graph"
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353:Ultrahomogeneity
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161:undirected graph
118:Strongly regular
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404:. The infinite
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154:Ludwig Schläfli
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476:complemented
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422:Gosset graph
400:, and the 5-
394:Turán graphs
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378:automorphism
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250:neighborhood
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203:that is the
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171:Construction
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146:graph theory
142:mathematical
139:
372:of at most
366:isomorphism
289:crown graph
126:Hamiltonian
693:Categories
506:References
406:Rado graph
255:isomorphic
240:GQ(2, 4).
205:complement
156:, is a 16-
114:Properties
675:MathWorld
386:connected
364:if every
316:◻
271:claw-free
144:field of
122:Claw-free
416:See also
185:polytope
75:Diameter
45:Vertices
625:2187738
536:Bibcode
261:of the
257:to the
158:regular
140:In the
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592:
148:, the
65:Radius
617:(PDF)
429:Notes
402:cycle
199:is a
99:51840
85:Girth
55:Edges
590:ISBN
338:and
248:The
209:skew
191:The
567:doi
544:doi
299:of
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695::
672:.
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621:MR
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563:28
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344:K
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329:K
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304:K
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