5-chromatic Strongly Regular Graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.

[1]  Andries E. Brouwer,et al.  Structure and uniqueness of the (81, 20, 1, 6) strongly regular graph , 1992, Discret. Math..

[2]  W. H. Haemers,et al.  Finite Geometry and Combinatorics: There exists no (76,21,2,7) strongly regular graph , 1993 .

[3]  C. Colbourn,et al.  CRC Handbook of Combinatorial Designs , 1996 .

[4]  J. H. Lint,et al.  Designs, graphs, codes, and their links , 1991 .

[5]  J. Seidel.  Strongly Regular Graphs with (—1, 1, 0) Adjacency Matrix Having Eigenvalue 3 , 1991 .

[6]  Nick C. Fiala.  Some topics in combinatorial design theory and algebraic graph theory , 2002 .

[7]  Edward Spence.  The Strongly Regular (40, 12, 2, 4) Graphs , 2000, Electron. J. Comb..

[8]  Andries E. Brouwer,et al.  Strongly Regular Graphs , 2022 .

[9]  E. V. Dam.  Three-Class Association Schemes , 1999 .

[10]  Alan J. Hoffman,et al.  On Moore Graphs with Diameters 2 and 3 , 1960, IBM J. Res. Dev..

[11]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[12]  Leonard H. Soicher,et al.  GRAPE: A System for Computing with Graphs and Groups , 1991, Groups And Computation.

[13]  A. Hoffman.  On the Polynomial of a Graph , 1963 .

[14]  吉岡 智晃.  Strongly Regular Graphs and Partial Geometries , 1993 .

[15]  J. J. Seidel,et al.  Strongly Regular Graphs Derived from Combinatorial Designs , 1970, Canadian Journal of Mathematics.

[16]  W. Haemers.  Interlacing eigenvalues and graphs , 1995 .

[17]  Peter C. Jurs,et al.  Mathematica , 2019, J. Chem. Inf. Comput. Sci..

[18]  Edward Spence,et al.  The Strongly Regular (45, 12, 3, 3) Graphs , 2006, Electron. J. Comb..

[19]  A.J.L. Paulus,et al.  Conference matrices and graphs of order 26 , 1973 .

[20]  Peter J. Cameron,et al.  Designs, graphs, codes, and their links , 1991 .

[21]  Andries E. Brouwer,et al.  Strongly regular graphs and partial geometries , 1984 .

[22]  R. Bellman,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[23]  Willem H. Haemers,et al.  The Search for Pseudo Orthogonal Latin Squares of Order Six , 2000, Des. Codes Cryptogr..

[24]  L. Beineke,et al.  Selected Topics in Graph Theory 2 , 1985 .

[25]  Vladimir D. Tonchev,et al.  Spreads in Strongly Regular Graphs , 1996, Des. Codes Cryptogr..

[26]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[27]  Ha Henny Wilbrink,et al.  A (57,14,1) strongly regular graph does not exist , 1978 .

[28]  S. Vanstone,et al.  Enumeration and design , 1984 .

[29]  Charles C. Sims,et al.  A simple group of order 44,352,000 , 1968 .

[30]  Allan Gewirtz.  SECTION OF MATHEMATICS: THE UNIQUENESS OF g(2,2,10,56)* , 1969 .

[31]  J. J. Seidel,et al.  Graphs and their spectra , 1989 .

[32]  Andries E. Brouwer,et al.  The Gewirtz Graph: An Exercise in the Theory of Graph Spectra , 1993, Eur. J. Comb..

[33]  Andries E. Brouwer.  The uniqueness of the strongly regular graph on 77 points , 1983, J. Graph Theory.

[34]  Willem H. Haemers,et al.  A (49, 16, 3, 6) Strongly Regular Graph Does Not Exist , 1989, Eur. J. Comb..

[35]  J. Seidel.  Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3 , 1968 .

[36]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[37]  A. Gewirtz,et al.  Graphs with Maximal Even Girth , 1969, Canadian Journal of Mathematics.

[38]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[39]  Mikhail H. Klin,et al.  A Root Graph that is Locally the Line Graph of the Peterson Graph , 2003, Discret. Math..

[40]  W. Haemers.  Eigenvalue techniques in design and graph theory , 1979 .

[41]  Brendan D. McKay,et al.  Classification of regular two-graphs on 36 and 38 vertices , 2001, Australas. J Comb..

[42]  Willem H. Haemers,et al.  The Pseudo-geometric Graphs for Generalized Quadrangles of Order (3, t) , 2001, Eur. J. Comb..

[43]  P. Cameron.  On groups with several doubly-transitive permutation representations , 1972 .