Siamese objects, and their relation to color graphs, association schemes and Steiner designs
The authors generalize the work of \textit{H. Kharaghani} and \textit{R. Torabi} [Graphs Comb. 19, 519--526 (2003; Zbl 1032.05008)] and study a class of edge partitions, called \textit{Siamese color graphs}, and its connections to a class of association schemes, called \textit{Siamese}. A Siamese color graph is an edge partition \(\{S,R_1,\dots,R_k\}\) of the complete graph \((V,V^2)\) that satisfies the following properties: {\parindent=5mm \begin{itemize}\item[1.]\((V,S)\) is the disjoint union of cliques of equal size; \item[2.]each \((V,R_i)\) is an imprimitive distance regular graph of diameter 3 with the antipodal system determined by \(S\); \item[3.]each \((V,S\cup R_i)\) is a strongly regular graph. \end{itemize}} An association scheme with classes \(S_1,\dots,S_m,R_1,\dots,R_k\) is called Siamese if \(\{\cup_j S_j,R_1,\dots,R_k\}\) is a Siamese color graph. The authors investigate relations of these objects to classical geometries, such as projective spaces over finite fields, finite generalized quadrangles, and construct few explicit examples for small \(| V| \). A more detailed account of this work is to appear elsewhere.
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