GRAPHS \(\Gamma\) OF DIAMETER 4 FOR WHICH \(\Gamma_{3,4}\) IS A STRONGLY REGULAR GRAPH WITH \(\mu=4,6\)

Abstract

We consider antipodal graphs \(\Gamma\) of diameter 4 for which \(\Gamma_{1,2}\) is a strongly regular graph. A.A. Makhnev and D.V. Paduchikh noticed that, in this case, \(\Delta=\Gamma_{3,4}\) is a strongly regular graph without triangles. It is known that in the cases \(\mu=\mu(\Delta)\in \{2,4,6\}\) there are infinite series of admissible parameters of strongly regular graphs with \(k(\Delta)=\mu(r+1)+r^2\), where \(r\) and \(s=-(\mu+r)\) are nonprincipal eigenvalues of \(\Delta\). This paper studies graphs with \(\mu(\Delta)=4\) and 6. In these cases, \(\Gamma\) has intersection arrays \(\{{r^2+4r+3},{r^2+4r},4,1;1,4,r^2+4r,r^2+4r+3\}\) and \(\{r^2+6r+5,r^2+6r,6,1;1,6,r^2+6r,r^2+6r+5\}\), respectively. It is proved that graphs with such intersection arrays do not exist.

Keywords

• distance-regular graph, strongly regular graph, triple intersection numbers