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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 = (q2m−1)/(q−1),
k = q(q2m−2−1)/(q−1),
λ = q2(q2m−4−1)/(q−1) +
q−1,
μ = (q2m−2−1)/(q−1),
The eigenvalues are
r = qm−1−1,
s = −qm−1−1.
Oε(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 = (q2m−1−1)/(q−1)
+ ε qm−1,
k = q(q2m−3−1)/(q−1)
+ ε qm−1,
k − λ − 1 = q2m−3,
μ = k/q,
The eigenvalues are εqm−1−1,
and −εqm−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 = (q2m−1)/(q−1),
k = q(q2m−2−1)/(q−1),
λ = q2(q2m−4−1)/(q−1)
+q−1,
μ = k/q = λ + 2,
The eigenvalues are
r = qm−1− 1,
s = −qm−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 = (qd−1)(qd+1/2+1)/(q−1),
k = q(qd−1−1)(qd−1/2+1)/(q−1),
λ = q2(qd−2−1)(qd−3/2+1)/(q−1)
+ q−1,
μ = k/q,
with eigenvalues
r = qd−1−1,
s = −qd−1/2−1.
If n is even, say n = 2d, then the parameters are
v = (qd−1)(qd−1/2+1)/(q−1),
k = q(qd−1−1)(qd−3/2+1)/(q−1),
λ = q2(qd−2−1)(qd−5/2+1)/(q−1)
+ q−1,
μ = k/q,
with eigenvalues
r = qd−1−1,
s = −qd−3/2−1.
E6(q)
For description and parameters of the E6(q) graphs, see BCN Table 10.8.
Here
v = (q12−1)(q9−1)/(q4−1)(q−1),
k = q(q3+1)(q8−1)/(q−1),
k − λ − 1 = q7(q5−1)/(q−1),
μ = (q3+1)(q4−1)/(q−1),
with eigenvalues
r = q8+q7+q6+
q5+q4−1,
s = −q3−1.
O10+(q) on one kind of maxes
The half dual polar space of type D5 is strongly regular.
The parameters are given in BCN 9.4.8 (as part of an infinite family
with increasing diameter). We have
v = (q4+1)(q3+1)(q2+1)(q+1)
k = q(q2+1)(q5−1)/(q−1),
k − λ − 1 = q5(q3−1)/(q−1),
μ = (q2+1)(q3−1)/(q−1),
with eigenvalues
r = q3(q3−1)/(q−1) −1,
s = −q2−1.
U5(q) on the totally isotropic lines
The dual polar graph of type 2A4 is strongly regular.
The parameters are given in BCN 9.4.3 (as part of an infinite family
with increasing diameter). We have
v = (q5/2+1)(q3/2+1),
k = q3/2(q+1),
k − λ − 1 = q5/2,
μ = q+1,
with eigenvalues
r = q3/2 − 1,
s = − q − 1.
O2mε(2) on the nonisotropic points
Nonisotropic points of O2mε(2), joined when on a tangent.
v = 22m−1 − ε2m−1,
k = 22m−2 − 1,
λ = 22m−3 − 2,
μ = 22m−3 + ε2m−2,
with eigenvalues ε2m−2 − 1 and −ε2m−1 − 1.
O2m+1(3) on one type of nonisotropic points
One type of nonisotropic points of O2m+1(3), joined when orthogonal
(i.e., connecting line is elliptic). There are two types of points x,
distinguished by the type ε of xperp.
v = 3m(3m+ε)/2,
k = 3m−1(3m−ε)/2,
λ = μ = 3m−1(3m−1−ε)/2,
with eigenvalues
r = 3m−1,
s = −3m−1.
O2mε(3) on one type of nonisotropic points
One type of nonisotropic points of O2mε(3),
joined when orthogonal (i.e., connecting line is elliptic).
v = 3m−1(3m−ε)/2,
k = 3m−1(3m−1−ε)/2,
λ = 3m−2(3m−1+ε)/2,
μ = 3m−1(3m−2−ε)/2,
with eigenvalues ε3m−1 and −ε3m−2.
U(n,q) on nonisotropic points
Let V be an n-dimensional vector space over the field GF(q2),
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 = qn−1(qn − ε)/(q + 1),
k = (qn−1 + ε)(qn−2 − ε),
λ = q2n−5(q+1) −
εqn−2(q−1) − 2,
μ = qn−3(q + 1)(qn−2 − ε),
with eigenvalues
εqn−2 − 1,
−ε(q2−q−1)qn−3 − 1.