By Jagdish Srivastava

ISBN-10: 0720422620

ISBN-13: 9780720422627

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**Extra resources for A Survey of Combinatorial Theory**

**Example text**

For the case m = 2, the parameters of a strongly regular graph may be said to characterize the graph if there is essentially one strongly regular graph with the given parameters. 1). He showed that the CH. 4 CHARACTERIZATION PROBLEMS OF COMBINATORIAL GRAPH THEORY 35 answer is in the affirmative if m > 8. This result can be translated into graph theoretic language. The line graph H of a graph G is defined to be a graph whose vertices correspond to the edges of G, and in which two vertices are adjacent or non-adjacent according as the corresponding edges of G have or do not have a common vertex.

6) This shows that G is a pseudo-net graph Gd(k). 5) is satisfied. Let GN and GN be the nets corresponding to the graphs G and G. They have the same k2 points. Through any point there pass r lines of GN (one belonging to each parallel class of GN) and k—r+1 lines of GN (one belonging to each parallel class of GN). If we now extend GN by adjoining the lines of GN then for the extended incidence structure Π there pass r+1 lines through every point. A line / of GN and a line I of GN cannot intersect in more than one point.

An example is provided by the lattice graph on 3 symbols. In Section 4 a further such graph, on 243 vertices, is constructed in three different ways. As a starting point for the constructions of this graph, in Section 3 the perfect ternary 2-error-correcting code is explained. This code has been t Current address: University of California, Berkeley, Calif. A. SEIDEL CH. 3 discovered by Golay [1949]. It was discussed by Coxeter [1958] in a geometric context, and by Bose [1961], who indicated its connection to the theory of confounding and fractional replication.

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