### 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.

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- Brouwer A.E., Cohen A.M., Neumaier A.
*Distance–Regular Graphs*. Berlin, Heidelberg: Springer–Verlag, 1989. 495 p. DOI: 10.1007/978-3-642-74341-2 - Coolsaet K., Jurišić A. Using equality in the Krein conditions to prove nonexistence of sertain distance-regular graphs.
*J. Combin. Theory Ser. A*, 2018. Vol. 115, No. 6. P. 1086–1095. DOI: 10.1016/j.jcta.2007.12.001 - Makhnev A.A., Paduchikh D.V. Inverse problems in the class of distance-regular graphs of diameter 4.
*Proc. Steklov Inst. Math.*, 2022. Vol. 317, No. Suppl. 1. P. S121–S129. DOI: 10.1134/S0081543822030105

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