A Survey of Combinatorial Theory by Jagdish Srivastava

A Survey of Combinatorial Theory by Jagdish Srivastava

By Jagdish Srivastava

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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 [1954], Bose and Mesner [1959]. 4. The triangular association scheme and the line graph of a complete graph We take an mxm square m ^ 3 and fill in the \m(m—\) positions above the leading diagonal by different treatments, taken in any order. The positions in the leading diagonal are left blank, while positions below this diagonal are filled so that the scheme is symmetrical with respect to the leading diagonal.

M. Goethals, and J. J. Seidel, 1970, Strongly regular graphs derived from combinatorial designs, Canad. J. Math. 22, 597-614. M. Golay, 1949, Notes on digital coding, Proc. E. 37, 637. A. J. Hoffman, and R. R. M. J. Res. Develop. 4, 497-504. A. J. Hoffman, 1963, On the polynomial of a graph, Am. Math. Monthly 70, 30-36. N. Robertson, (private communication). J. J. Seidel, 1968, Strongly regular graphs with (—1,1,0) adjacency matrix having eigen­ value 3, Linear Algebra Appl. 1, 281-298. J. J. Seidel, 1969, Strongly regular graphs, Recent Progress in Combinatorics (W'.

The question is open for 9 < m ^ 16 though no exceptional cases are known. For q > 3, nothing is known about the case when m ^ 2<7(<7-l) + 4. (C) A cubic lattice graph of order m is a graph G whose vertices can be identified with the ordered triplets on m symbols so that two vertices are adjacent if the corresponding triplets have common symbols in exactly two positions. If G is a cubic lattice graph of order m then G possesses the following properties: (c0) G has v = m3 vertices. (cx) G is regular of degree 3(ra— 1).

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