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# SRG family parameters

Below we give the parameters for various infinite families of
strongly regular graphs with a nice group.
## Lines in PG(d−1,q)

Take the lines in PG(d−1,q), adjacent when they meet in a point.
This is a Grassmann graph of diameter 2.
(For the parameters, see there.)
## O(2m+1,q) - Points on a quadric in PG(2m,q)

Take the points on a nondegenerate quadric in PG(2m,q), adjacent when
orthogonal. This graph is strongly regular with parameters

*v* = (*q*^{2m}−1)/(*q*−1),

*k* = *q*(*q*^{2m−2}−1)/(*q*−1),

λ = *q*^{2}(*q*^{2m−4}−1)/(*q*−1) +
*q*−1,

μ = (*q*^{2m−2}−1)/(*q*−1),
The eigenvalues are

*r* = *q*^{m−1}−1,

*s* = −*q*^{m−1}−1.

## O^{ε}(2m,q) - Points on a quadric in PG(2m−1,q)

Take the points on a nondegenerate quadric in PG(2m−1,q), hyperbolic
if ε = 1, elliptic if ε = −1, where the points are adjacent when
orthogonal. This graph is strongly regular with parameters

*v* = (*q*^{2m−1}−1)/(*q*−1)
+ ε *q*^{m−1},

*k* = q(*q*^{2m−3}−1)/(*q*−1)
+ ε *q*^{m−1},

*k* − λ − 1 = *q*^{2m−3},

μ = *k*/*q*,
The eigenvalues are ε*q*^{m−1}−1,
and −ε*q*^{m−2}−1.

## Sp(2m,q)

Let V be a 2m-dimensional vector space over the field GF(*q*),
provided with a nondegenerate symplectic form.
Take the points of PV, adjacent when orthogonal.
This graph is strongly regular with parameters

*v* = (*q*^{2m}−1)/(*q*−1),

*k* = *q*(*q*^{2m−2}−1)/(*q*−1),

λ = *q*^{2}(*q*^{2m−4}−1)/(*q*−1)
+*q*−1,

μ = *k*/*q* = λ + 2,
The eigenvalues are

*r* = *q*^{m−1}− 1,

*s* = −*q*^{m−1}− 1.

## U(n,q)

Let V be an n-dimensional vector space over the field GF(*q*),
where *q* is a square, provided with a nondegenerate
Hermitean form.
Take the isotropic points of PV, adjacent when orthogonal.
This graph is strongly regular.
If n is odd, say n = 2d+1, then the parameters are

*v* = (*q*^{d}−1)(*q*^{d+1/2}+1)/(*q*−1),

*k* = *q*(*q*^{d−1}−1)(*q*^{d−1/2}+1)/(*q*−1),

λ = *q*^{2}(*q*^{d−2}−1)(*q*^{d−3/2}+1)/(*q*−1)
+ *q*−1,

μ = *k*/*q*,

with eigenvalues

*r* = *q*^{d−1}−1,

*s* = −*q*^{d−1/2}−1.

If n is even, say n = 2d, then the parameters are

*v* = (*q*^{d}−1)(*q*^{d−1/2}+1)/(*q*−1),

*k* = *q*(*q*^{d−1}−1)(*q*^{d−3/2}+1)/(*q*−1),

λ = *q*^{2}(*q*^{d−2}−1)(*q*^{d−5/2}+1)/(*q*−1)
+ *q*−1,

μ = *k*/*q*,

with eigenvalues

*r* = *q*^{d−1}−1,

*s* = −*q*^{d−3/2}−1.

## E_{6}(q)

For description and parameters of the E_{6}(q) graphs, see BCN Table 10.8.
Here

*v* = (*q*^{12}−1)(*q*^{9}−1)/(*q*^{4}−1)(*q*−1),

*k* = *q*(*q*^{3}+1)(*q*^{8}−1)/(*q*−1),

*k* − λ − 1 = *q*^{7}(*q*^{5}−1)/(*q*−1),

μ = (*q*^{3}+1)(*q*^{4}−1)/(*q*−1),
with eigenvalues

*r* = *q*^{8}+*q*^{7}+*q*^{6}+
*q*^{5}+*q*^{4}−1,

*s* = −*q*^{3}−1.

## O_{10}^{+}(q) on one kind of maxes

The half dual polar space of type D_{5} is strongly regular.
The parameters are given in BCN 9.4.8 (as part of an infinite family
with increasing diameter). We have

*v* = (*q*^{4}+1)(*q*^{3}+1)(*q*^{2}+1)(*q*+1)

*k* = *q*(*q*^{2}+1)(*q*^{5}−1)/(*q*−1),

*k* − λ − 1 = *q*^{5}(*q*^{3}−1)/(*q*−1),

μ = (*q*^{2}+1)(*q*^{3}−1)/(*q*−1),
with eigenvalues

*r* = *q*^{3}(*q*^{3}−1)/(*q*−1) −1,

*s* = −*q*^{2}−1.

## U_{5}(q) on the totally isotropic lines

The dual polar graph of type ^{2}A_{4} is strongly regular.
The parameters are given in BCN 9.4.3 (as part of an infinite family
with increasing diameter). We have

*v* = (*q*^{5/2}+1)(*q*^{3/2}+1),

*k* = *q*^{3/2}(*q*+1),

*k* − λ − 1 = *q*^{5/2},

μ = *q*+1,
with eigenvalues

*r* = *q*^{3/2} − 1,

*s* = − *q* − 1.

## O_{2m}^{ε}(2) on the nonisotropic points

Nonisotropic points of O_{2m}^{ε}(2), joined when on a tangent.

*v* = 2^{2m−1} − ε2^{m−1},

*k* = 2^{2m−2} − 1,

λ = 2^{2m−3} − 2,

μ = 2^{2m−3} + ε2^{m−2},

with eigenvalues ε2^{m−2} − 1 and −ε2^{m−1} − 1.
## O_{2m+1}(3) on one type of nonisotropic points

One type of nonisotropic points of O_{2m+1}(3), joined when orthogonal
(i.e., connecting line is elliptic). There are two types of points x,
distinguished by the type ε of x^{perp}.

*v* = 3^{m}(3^{m}+ε)/2,

*k* = 3^{m−1}(3^{m}−ε)/2,

λ = μ = 3^{m−1}(3^{m−1}−ε)/2,
with eigenvalues

*r* = 3^{m−1},

*s* = −3^{m−1}.

## O_{2m}^{ε}(3) on one type of nonisotropic points

One type of nonisotropic points of O_{2m}^{ε}(3),
joined when orthogonal (i.e., connecting line is elliptic).

*v* = 3^{m−1}(3^{m}−ε)/2,

*k* = 3^{m−1}(3^{m−1}−ε)/2,

λ = 3^{m−2}(3^{m−1}+ε)/2,

μ = 3^{m−1}(3^{m−2}−ε)/2,
with eigenvalues ε3^{m−1} and −ε3^{m−2}.

## U(n,q) on nonisotropic points

Let V be an n-dimensional vector space over the field GF(*q*^{2}),
provided with a nondegenerate Hermitean form.
Take the nonisotropic points of PV, adjacent when joined by a tangent.
This graph is strongly regular. Let ε = (−1)^{n}.
Then the parameters are

*v* = *q*^{n−1}(*q*^{n} − ε)/(*q* + 1),

*k* = (*q*^{n−1} + ε)(*q*^{n−2} − ε),

λ = *q*^{2n−5}(*q*+1) −
ε*q*^{n−2}(*q*−1) − 2,

μ = *q*^{n−3}(*q* + 1)(*q*^{n−2} − ε),
with eigenvalues
ε*q*^{n−2} − 1,
−ε(*q*^{2}−*q*−1)*q*^{n−3} − 1.