By Jagdish Srivastava
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11. , Xu are dependent. Therefore, these vectors and the zero vector represent all vectors of V(5, 3). Now consider the 3 5 vectors of V(5, 3) as the vertices of a graph. Any two vertices are called adjacent if the difference of the corresponding vectors is one of ±xl9 ±x2,.. ·, ± * n · Again, the triangle and the quadrangle property are easily verified. Finally, we indicate a third construction of the 243-graph, similar to the construction of the 2048-graph obtained from the (24, 12) binary Golay code in Goethals and Seidel .
If P is a vertex of G not contained in a clique Kv of Σ, there are exactly / vertices of Kx which are adjacent to G. Conversely, if there exists a graph with a set of cliques Σ satisfying A*l-A*4, then it is isomorphic to a partial geometry (r, k, t), the vertices of the graph CH. 4 CHARACTERIZATION PROBLEMS OF COMBINATORIAL GRAPH THEORY 43 being the points of the geometry and the cliques being the lines of the geo metry. The containing contained relation in G is the incidence relation in the geometry.
3) hold. The concept of a strongly regular graph was first introduced by Bose . In what follows we shall assume that p\x > 0. Then a strongly regular graph is a connected graph. 5) where / is the unit matrix of order v, and J is a v x v matrix with each element unity. 34 CH. 4 R. C. BOSE nx is a simple eigenvalue of ^4. There are only two other distinct eigenvalues 0! and 02 with multiplicities OL1 and a2, given by αι ? 8) A = y + 2J8+1. 6) are integral. For proofs of these results see Connor and Clatworthy , Bose and Mesner .
A Survey of Combinatorial Theory by Jagdish Srivastava