On Strongly Regular Self- Complementary

Sergio Ruiz Graphs’

UNI VERSIDA D CA TOLICA D E VAL PA RAISO VAL PA RAISO, CHILE

ABSTRACT

It is shown that certain conditions assumed on a regular self-complementary graph are not sufficient for the graph to be strongly regular, answering in the negative a question posed by Kotzig in [ l ] .

In this Note we solve problem 5 presented by Anton Kotzig in [ 11: “Is it true that a regular self-complementary graph G is strongly regular if and only if F ( G ) = V ( G ) and N(G) = E(G)?” Before proving that the answer is negative we provide the definitions and results used. We take Harary’s definition of a graph G; V( G) and E( G) are the vertex set and the edge set of G. The complement G of a graph G is the graph with vertex set V( G). Two vertices are adjacent in G if and only if these vertices are not adjacent in G. A graph G is self-complementary if G and G are isomorphic. Any isomorphism z between G and G is called by Kotzig an isomorphism-permutation. It is known that the cycle structure of z for a regular self-complementary graph consists of a product of cycles of length equal to a multiple of four and exactly one cycle of length one. The subset F( G) of V( G) consists of those vertices which are fixed by at least one isomorphism-permutation of the regular self- complementary graph G. The subset N( G) of E( G) consists of those edges (u , v) of G with the property that there is at least one isomorphism- permutation P such that u and v are neighbors in a cycle of P.

A regular graph G is strongly regular if every edge belongs to the same number d of triangles C,, and, for each pair u, v of nonadjacent vertices exactly c other vertices are simultaneously joined to u and v.

As an example of these concepts we must mention the so called Paley graph P(q) , where q is a prime power such that q 1 (mod 4): it has as vertex set the elements of GF(q) , and two vertices are adjacent if and only if

‘Research supported by Direccion General de Investigacion de la Universidad Catolica de Valparaiso under the project 01 1-04-78.

Journal of Graph Theory, Vol. 5 (1 981 ) 21 3-21 5 o 1981 by John Wiley& Sons., Inc. CCCO364-9024/81/020213-03$01 .OO

214 JOURNAL OF GRAPH THEORY

12

7 8

FIGURE 1

their difference is a nonzero square. This graph is strongly regular and self- complementary.

Returning to the original problem, we want to exhibit a graph G that is regular and self-complementary, satisfies F( G) = V( G) and N( G) = E( G), but is not strongly regular. Such a graph is shown in Figure 1. It was found by the following construction which produces all the self-complemenatry vertex- transitive graphs with a prime number of vertices.

For any prime p of the form 4k 4- 1, take a subset S of the multiplicative group Z:, with k elements, satisfying

for some u in 2:. Then the graph G = G ( S , p ) has as vertices the elements of 2,. Two vertices x and y are adjacent if and only if x f y and y = hx for some h in S .

Figure 1 shows G ( S , 13) for S = (1, 3, 6). The graph is, of course, circulant. This guarantees that the automorphism group induces at most three orbits of edges. Clearly edges of the same length in the drawing belong to the same orbit. But edges of different lengths lie on different numbers of triangles. Thus there are exactly three orbits. Because of the vertex transitivity it is immediate that F( G) = V( G).

SELF-COMPLEMENTARY GRAPHS 21 5

Now let us consider the isomorphism-permutation z = (1 2 3 4) (5 6 7 8 ) ( 9 10 1 1 12) (1 3) of G. Three representatives of each orbit of edges are ( 5 , 6 } , (3,4} and (1 1, 10). The end vertices of each edge are neighbors in a cycle of z. Using transitivity in each orbit we conclude that N( G) = E( G ) .

We note that the graph in Figure 1 and the Paley graphP( 13) = G(S, 13) for S = f 1, 3, 4) are the only vertex-transitive, self-complementary graphs with 13 vertices.

ACKNOWLEDGMENTS

The author wishes to express his gratitude to David Powers for his help in translation of this note into English and to Wadim Praus for his cooperation on the drawing of Figure 1.

References

[ 11 A. Kotzig, Selected open problems in graph theory. In Graph Theoty

[2] S . Ruiz, Deterrninacion de 10s grafos autocomplementarios vertice- and Related Topics, Academic, New York (1979).

transitivos con un numero primo de puntos. Submitted.