In a previous paper it was established that there are at least 27 non-isomorphic strongly regular (40, 12, 2, 4) graphs. Using a different and more efficient method we have re-investigated these graphs and have now been able to determine them all, and so complete the classification. We have discovered that there are precisely 28 non-isomorphic (40, 12, 2, 4) strongly regular graphs. The one tha...

Article history: Received 24 August 2010 Available online 8 November 2011

Let ∆ be one of the dual polar spaces DQ(8, q), DQ−(7, q), and let e : ∆ → Σ denote the spin-embedding of ∆. We show that e(∆) is a two-intersection set of the projective space Σ. Moreover, if ∆ ∼= DQ−(7, q), then e(∆) is a (q + 1)-tight set of a nonsingular hyperbolic quadric Q(7, q) of Σ ∼= PG(7, q). This (q + 1)-tight set gives rise to more examples of (q + 1)-tight sets of hyperbolic quadri...

Let Γ be a distance-regular graph with diameter d and Kneser graph K = Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We prov...

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations. Perhaps the most important subclass of these objects is the class of cometric Q-antipodal association schemes. The concept of a cometric association scheme (the dual version of a distance-regular graph) is well-known; however, until recently it has not been studied well outside the ar...

We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint un...

Our general investigation of universal algebras obtained from multialgebras via strongly regular equivalence relations provides useful general results concerning fuzzy set topics related to multialgebra theory. We also give many hints on how to connect our approach with the results from the literature. Acknowledgement. The first author was supported by the CNCS-UEFISCDI grant PNII-RU-TE-2011-3-...

We present new abelian partial difference sets and amorphic group schemes of both Latin square type and negative Latin square type in certain abelian p-groups. Our method is to construct what we call pseudo-quadratic bent functions and use them in place of quadratic forms. We also discuss a connection between strongly regular bent functions and amorphic group schemes.