N.D. Zyulyarkina, M.Kh. Shermetova. Large vertex-symmetric Higman graphs with $\mu=6$ ... P. 146-155

Full text (in Russian)

A strongly regular graph with $v={m\choose 2}$ and $k=2(m-2)$ is called a Higman graph. In such a graph, the parameter $\mu$ is 4, 6, 7, or 8. If $\mu=6$, then $m\in\{9,17,27,57\}$. Vertex-symmetric Higman graphs were classified by N.D. Zyulyarkina and A.A. Makhnev (all of these graphs turned out to have rank 3). A program of classification of vertex-symmetric Higman graphs is implemented. Earlier Zyulyarkina and Makhnev found vertex-symmetric Higman graphs with $\mu=6$ and $m\in\{9,17\}$. In the present paper, vertex-symmetric Higman graphs with $\mu=6$ and $m\in{27,57}$ are studied. It is interesting that the group $G/S(G)$ may contain two components $L$ and $M$. In the case $m=27$, we have $M\cong A_5,A_6$ and $L\cong L_3(3)$; in the case $m=57$, we have either $M\cong PSp_4(3)$ and $L\cong L_3(7)$ or $M\cong A_6$ and $L\cong J_1$.

Keywords: distance-regular graph, graph automorphism

Received February 20, 2018

Revised October 16, 2018

Accepted October 22, 2018

Natal’ya Dmitrievna Zyulyarkina, Dr. Phys.-Math. Sci., Prof., South Ural State University, Chelyabinsk, 454080 Russia, e-mail: toddeath@yandex.ru

Mariyana Khusenovna Shermetova, doctoral student, Kabardino-Balkarian State University named after H.M.Berbekov, Nal’chik, 360004 Russia, e-mail: mariyana1992@mail.ru

REFERENCES

1.   Higman D.G. Characterization of families of rank 3 permutation groups by the subdegrees, I. Arch. Math., 1970, vol. 21, no. 1, pp. 151–156. doi: 10.1007/BF01220896

2.   Zyulyarkina N.D., Makhnev A.A. Edge-symmetric semitriangular Higman graphs. Dokl. Math., 2014, vol. 90, no. 3, pp. 701–705. doi: 10.1134/S1064562414070199

3.   Zyulyarkina N.D., Makhnev A.A., Paduchikh D.V., Khamgokova M. M. Vertex-transitive semi-triangular graphs with $\mu=7$. Sib. Elektron. Mat. Izv., 2017, vol. 14, pp. 1198–1206 (in Russian). doi: 10.17377/semi.2017.14.101

4.   Zyulyarkina N.D., Makhnev A.A. Small vertex-symmetric Higman graphs with $\mu=6$. Sib. Elektron. Mat. Izv., 2018, vol. 15, pp. 54–59 (in Russian). doi: 10.17377/semi.2018.15.007

5.   Zyulyarkina N.D., Makhnev A.A. Automorphisms of semitriangular graphs with $\mu=6$. Dokl. Math., 2009, vol. 79, no. 3, pp. 373–376. doi: 10.1134/S106456240903020X

6.   Cameron P. Permutation Groups. London: Cambridge Univ. Press, 1999, 220 p. ISBN: 0-521-65302-9 .

7.   Behbahani M., Lam C. Strongly regular graphs with nontrivial automorphisms. Discrete Math., 2011, vol. 311, no. 2-3, pp. 132–144. doi: 10.1016/j.disc.2010.10.005

8.   Haemers W.H. Interlacing eigenvalues and graphs. Linear Algebra Appl., 1995, vol. 226–228, pp. 593–616. doi: 10.1016/0024-3795(95)00199-2

9.   Macay M., Siran J. Search for properties of the missing Moore graph. Linear Algebra Appl., 2010, vol. 432, no. 9, pp. 2381–2398. doi: 10.1016/j.laa.2009.07.018

10.   Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Sib. Elektron. Mat. Izv., 2009, vol. 6, pp. 1–12.