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 ).