1-Homogeneous Graphs with Cocktail Party μ -Graphs
Aleksandar Jurišić
and Jack Koolen
DOI: 10.1023/A:1025133213542
Abstract
Let 
be a graph with diameter d 
2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition
be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition {{ z 
V( ) | 
( z, y) = i, ( x, z) = j} | 0 
i, j d}
V( ) | ( z, y) = i, ( x, z) = j} | 0 i, j d}Pages: 79–98
Keywords: distance-regular graph; 1-homogeneous; cocktail party graph; Johnson graph
Full Text: PDF
References
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11. A. Juri\check sić and J. Koolen, “A local approach to 1-Homogeneous graphs,” Designs, Codes and Cryptography 21 (2000), 127-147.
12. A. Juri\check sić and J. Koolen, “Krein parameters and antipodal tight graphs with diameter 3 and 4,” Discr. Math. 244 (2002), 181-202.
13. W.J. Martin, “Completely regular codes,” Ph.D. Thesis, University of Waterloo, 1992.
14. A. Neumaier, “Completely regular codes,” Discrete Math. 106/107 (1992), 353-360.
15. A. Neumaier, “Characterization of a class of distance regular graphs,” J. Reine Angew. Math. 357 (1985), 182-192.
16. K. Nomura, “Homogeneous graphs and regular near polygons,” J. Combin. Theory Ser. B 60 (1994), 63-71.
17. J.J. Seidel, “Strongly-regular graphs with (1,-1,0) adjacency matrix having eigenvalue 3,” Linear Alg. and Appl. 1 (1968), 281-298. JURI \check SI Ć AND KOOLEN
18. H.C.A. van Tilborg, “Uniformly packed codes,” Ph.D. Thesis, Eindhoven University of Technology, 1976.
19. P. Terwilliger, “Distance-regular graphs with girth 3 or 4, I,” J. Combin. Th. (B) 39 (1985), 265-281.
20. P. Terwilliger, “The Johnson graph J (d, r ) is unique if (d, r ) = (2, 8),” Discrete Math. 58 (1986), 175- 189.
21. P. Terwilliger, “Root systems and the Johnson and Hamming graphs,” European J. Combin. 8 (1987), 73- 102.
22. J.A. Thas, “Extensions of finite generalized quadrangles,” Symposia Mathematica, Vol. XXVIII (Rome 1983), Academic Press, London 1986, pp. 127-143.
2. A.E. Brouwer, “On the uniqueness of a certain thin near octagon (or partial 2-geometry, or parallelism) derived from the binary Golay code,” IEEE Trans. Inform. Theory 29(3) (1983), 370-371.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
4. P.J. Cameron, “Parallelisms of complete designs,” London Math. Soc. Lecture Notes 23, Cambridge Univ. Press, Cambridge (1976).
5. P.J. Cameron, D.R. Hughes, and A. Pasini, “Extended generalized quadrangles,” Geometriae Dedicata 35 (1990), 193-228.
6. Ph. Delsarte, “An algebraic approach to the association schemes of coding theory,” Phillips Research Reports Suppl. 10 (1973).
7. A. Gardiner, “Antipodal covering graphs,” J. Combin. Th. (B) 16 (1974), 255-273.
8. C.D. Godsil, Algebraic combinatorics, Chapman and Hall, New York (1993).
9. A. Juri\check sić and J. Koolen, “Nonexistence of some antipodal distance-regular graphs of diameter four,” Europ. J. Combin. 21 (2000), 1039-1046.
10. A. Juri\check sić, J. Koolen, and P. Terwilliger, “Tight Distance-Regular Graphs,” J. Alg. Combin. 12 (2000), 163-197.
11. A. Juri\check sić and J. Koolen, “A local approach to 1-Homogeneous graphs,” Designs, Codes and Cryptography 21 (2000), 127-147.
12. A. Juri\check sić and J. Koolen, “Krein parameters and antipodal tight graphs with diameter 3 and 4,” Discr. Math. 244 (2002), 181-202.
13. W.J. Martin, “Completely regular codes,” Ph.D. Thesis, University of Waterloo, 1992.
14. A. Neumaier, “Completely regular codes,” Discrete Math. 106/107 (1992), 353-360.
15. A. Neumaier, “Characterization of a class of distance regular graphs,” J. Reine Angew. Math. 357 (1985), 182-192.
16. K. Nomura, “Homogeneous graphs and regular near polygons,” J. Combin. Theory Ser. B 60 (1994), 63-71.
17. J.J. Seidel, “Strongly-regular graphs with (1,-1,0) adjacency matrix having eigenvalue 3,” Linear Alg. and Appl. 1 (1968), 281-298. JURI \check SI Ć AND KOOLEN
18. H.C.A. van Tilborg, “Uniformly packed codes,” Ph.D. Thesis, Eindhoven University of Technology, 1976.
19. P. Terwilliger, “Distance-regular graphs with girth 3 or 4, I,” J. Combin. Th. (B) 39 (1985), 265-281.
20. P. Terwilliger, “The Johnson graph J (d, r ) is unique if (d, r ) = (2, 8),” Discrete Math. 58 (1986), 175- 189.
21. P. Terwilliger, “Root systems and the Johnson and Hamming graphs,” European J. Combin. 8 (1987), 73- 102.
22. J.A. Thas, “Extensions of finite generalized quadrangles,” Symposia Mathematica, Vol. XXVIII (Rome 1983), Academic Press, London 1986, pp. 127-143.
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