On graphs whose vertex neighborhoods are locally pseudocyclic graphs
V. Kabanov, A. Makhnev
A locally pseudocyclic graph is a strongly regular graph with \(\lambda=2\). We study
the distance-regular graphs whose vertex neighborhoods are locally pseudocyclic graphs.
In [1], a program is proposed for the study of distance-regular graphs whose vertex neighborhoods
are strongly regular graphs with eigenvalue 3. In [1], there is also a reduction of the problem to the case where the
vertex neighborhoods belong to a finite set of exceptional graphs.
In the present work, we study completely regular graphs whose vertex neighborhoods are locally pseudocyclic graphs with eigenvalue 3.
Theorem. Let \(\Gamma\) be a completely regular graph whose vertex neighborhoods are locally pseudocyclic graphs
with nonprincipal eigenvalue \(3\). Then either \(d(\Gamma)\ge 4\), the vertex neighborhoods are strongly regular graphs with parameters \((96,19,2,4)\) and
\(\mu=12,16\), or \(d(\Gamma)=3\) and one of the following holds:
\((1)\) the vertex neighborhoods are strongly regular graphs with parameters \((96,19,2,4)\) and \(\mu=12,16,19,24,32\);
\((2)\) the vertex neighborhoods are strongly regular graphs with parameters \((196,39,2,9)\) and \(\mu=42,48,52,56\);
\((3)\) the vertex neighborhoods are strongly regular graphs with parameters \((256,51,2,12)\) and \(\mu=48,51,64,68,96\).
Corollary.A distance-regular graphs whose vertex neighborhoods are pseudocyclic graphs with eigenvalue
\(3\) has intersection array \(\{96,76,1;1,19,96\}\) or \(\{256,204,1;1,51,256\}\).
Acknowledgement. The work is partially supported by RFBR (grants 12-01-00012), RFBR – NSFC of China (grant 12-01-91155), by the Program of the Department of Mathematical Sciences of RAS (project 12-T-1-1003), and by the Joint Research Program of UB RAS with SB RAS (project 12-C-1-1018) and with NAS of Belarus (project 12-C-1-1009).
References
À. Makhnev, On strongly regular graphs with eigenvalue 3 and their extensions, Dokl. Akad. Nauk, 451, N5 (2013), 475–478. (Russian)
