Geodesic
Antipodal Krein graphs and distance-regular graphs close to them
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 54-57.

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An antipodal nonbipartite distance-regular graph $\Gamma$ of diameter 3 has an intersection array $\{k(r-1)c_2,1;1,c_2,k\}$ ($c_2$) and eigenvalues $k,n,-1$, and $-m$, where $n$ and $-m$ are the roots of the quadratic equation $x^2-(a_1-c_2)x-k=0$. The Krein bound $q^3_{33}\geq0$ gives $m\leq n^2$ if $r\ne2$. In the case $m=n^2$, following Godsil, we call $\Gamma$ an antipodal Krein graph. The point graph $\Sigma$ of $GQ(q,q^2)$ having spread gives an antipodal Krein graph with $r=q+1$. If $\Sigma$ has an automorphism $\sigma$ of order $f$ that fixes every component of the spread, then the graph $\overline\Sigma=\Sigma/\langle\sigma\rangle$ whose vertices are $\sigma$-orbits on a point set and two orbits are adjacent if a vertex of one orbit is adjacent to a vertex of the other is a distance-regular graph with intersection array $\{q^3,((q+1)/(f-1)(q^2-1)f,1;1,(q^2-1)f,q^3\}$ and every local subgraph $\Delta(u)$ is pseudogeometric for $pG_{f-1}(q-1,(q+1)(f-1))$. If $f=2$, then we have a pseudogeometric graph for $GQ(q-1,q+1)$. Hence, a locally pseudo $GQ(4,6)$ graph with intersection array $\{125,96,1;1,48,125\}$ and a locally pseudo $GQ(6,8)$ graph with intersection array $\{343,288,1;1,96,343\}$ exist.
Keywords: distance-regular graph, antipodal Krein graph. 
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     author = {A. A. Makhnev},
     title = {Antipodal {Krein} graphs and distance-regular graphs close to them},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {54--57},
     publisher = {mathdoc},
     volume = {492},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_492_a10/}
}
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A. A. Makhnev. Antipodal Krein graphs and distance-regular graphs close to them. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 54-57. http://geodesic.mathdoc.fr/item/DANMA_2020_492_a10/

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