Strongly Regular Graphs Having Strongly Regular Subconstituents

The chapter reviews the Bose–Mesner algebra of a strongly regular graph Γ , and presents in the i th eigenspace V i , i ∈ {1, 2} a special basis relative to some vertex of the graph. The Krein parameter is related to the components, with respect to the special basis, of a symmetric 3-tensor. The chapter also presents a definition of spherical i -designs in terms of tensors. The theorems explained in the chapter show that q i ii = 0 if and only if Γ yields a spherical 3-design in V i , and if and only if the sub-constituents of Γ yield spherical 2-designs in a hyperplane of V i . The chapter presents a comparison of the restricted spectra of the sub-constituents of Γ . The chapter focuses on Smith graphs having (q + l) (q 3 + 1) vertices, and on those having 16, 27, 100, 112, 162, and 275 vertices. Any graph of the first family is the point graph of a generalized quadrangle ( q , q 2 ).

[1]  Some Properties of Character Products , 1977 .

[2]  J. J. Seidel,et al.  A SURVEY OF TWO-GRAPHS , 1976 .

[3]  Xavier L. Hubaut,et al.  Strongly regular graphs , 1975, Discret. Math..

[4]  Philippe Delsarte,et al.  Four Fundamental Parameters of a Code and Their Combinatorial Significance , 1973, Inf. Control..

[5]  J. Tits.  Ovoßdes et Groupes de Suzuki , 1962 .

[6]  J. H. Lint,et al.  Graph theory, coding theory, and block designs , 1975 .

[7]  R. C. Bose,et al.  On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs , 1959 .

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

[9]  D. G. Higman.  Invariant Relations, Coherent Configurations and Generalized Polygons , 1975 .

[10]  J. J. Seidel,et al.  Orthogonal Matrices with Zero Diagonal , 1967, Canadian Journal of Mathematics.

[11]  D. Corneil,et al.  An Efficient Algorithm for Graph Isomorphism , 1970, JACM.

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

[13]  D. G. Higman.  Coherent configurations , 1975 .

[14]  S. S. Shrikhande,et al.  The Uniqueness of the $\mathrm{L}_2$ Association Scheme , 1959 .

[15]  J. J. Seidel,et al.  The regular two-graph on 276 vertices , 1975, Discret. Math..

[16]  R. C. Bose,et al.  Geometric and pseudo-geometric graphs (q2+1,q+1,1) , 1972 .

[17]  R. C. Bose.  Strongly regular graphs, partial geometries and partially balanced designs. , 1963 .

[18]  C. R. Rao,et al.  Factorial Experiments Derivable from Combinatorial Arrangements of Arrays , 1947 .

[19]  M. S. Smith,et al.  On rank 3 permutation groups , 1975 .

[20]  D. G. Higman.  Finite permutation groups of rank 3 , 1964 .

[21]  J. Thas.  4-Gonal configurations with parameters r=q2+1 and k=q+1 , 1974 .

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

[23]  R. H. Bruck.  Finite nets. II. Uniqueness and imbedding. , 1963 .

[24]  E. Witt.  über Steinersche Systeme , 1937 .