Families of graphs derived from classical geometries over finite fields¶
These include graphs of polar spaces, affine polar graphs, graphs related to Hermitean unitals, graphs on nonisotropic points, etc
The methods defined here appear in sage.graphs.graph_generators
.
- sage.graphs.generators.classical_geometries.AffineOrthogonalPolarGraph(d, q, sign='+')¶
Returns the affine polar graph \(VO^+(d,q),VO^-(d,q)\) or \(VO(d,q)\).
Affine Polar graphs are built from a \(d\)-dimensional vector space over \(F_q\), and a quadratic form which is hyperbolic, elliptic or parabolic according to the value of
sign
.Note that \(VO^+(d,q),VO^-(d,q)\) are strongly regular graphs, while \(VO(d,q)\) is not.
For more information on Affine Polar graphs, see Affine Polar Graphs page of Andries Brouwer’s website.
INPUT:
d
(integer) –d
must be even ifsign is not None
, and odd otherwise.q
(integer) – a power of a prime number, as \(F_q\) must exist.sign
– must be equal to"+"
,"-"
, orNone
to compute (respectively) \(VO^+(d,q),VO^-(d,q)\) or \(VO(d,q)\). By defaultsign="+"
.
Note
The graph \(VO^\epsilon(d,q)\) is the graph induced by the non-neighbors of a vertex in an
Orthogonal Polar Graph
\(O^\epsilon(d+2,q)\).EXAMPLES:
The
Brouwer-Haemers graph
is isomorphic to \(VO^-(4,3)\):sage: g = graphs.AffineOrthogonalPolarGraph(4,3,"-") sage: g.is_isomorphic(graphs.BrouwerHaemersGraph()) True
Some examples from Brouwer’s table or strongly regular graphs:
sage: g = graphs.AffineOrthogonalPolarGraph(6,2,"-"); g Affine Polar Graph VO^-(6,2): Graph on 64 vertices sage: g.is_strongly_regular(parameters=True) (64, 27, 10, 12) sage: g = graphs.AffineOrthogonalPolarGraph(6,2,"+"); g Affine Polar Graph VO^+(6,2): Graph on 64 vertices sage: g.is_strongly_regular(parameters=True) (64, 35, 18, 20)
When
sign is None
:sage: g = graphs.AffineOrthogonalPolarGraph(5,2,None); g Affine Polar Graph VO^-(5,2): Graph on 32 vertices sage: g.is_strongly_regular(parameters=True) False sage: g.is_regular() True sage: g.is_vertex_transitive() True
- sage.graphs.generators.classical_geometries.AhrensSzekeresGeneralizedQuadrangleGraph(q, dual=False)¶
Return the collinearity graph of the generalized quadrangle \(AS(q)\), or of its dual
Let \(q\) be an odd prime power. \(AS(q)\) is a generalized quadrangle (Wikipedia article Generalized_quadrangle) of order \((q-1,q+1)\), see 3.1.5 in [PT2009]. Its points are elements of \(F_q^3\), and lines are sets of size \(q\) of the form
\(\{ (\sigma, a, b) \mid \sigma\in F_q \}\)
\(\{ (a, \sigma, b) \mid \sigma\in F_q \}\)
\(\{ (c \sigma^2 - b \sigma + a, -2 c \sigma + b, \sigma) \mid \sigma\in F_q \}\),
where \(a\), \(b\), \(c\) are arbitrary elements of \(F_q\).
INPUT:
q
– a power of an odd prime numberdual
– ifFalse
(default), return the collinearity graph of \(AS(q)\). Otherwise return the collinearity graph of the dual \(AS(q)\).
EXAMPLES:
sage: g=graphs.AhrensSzekeresGeneralizedQuadrangleGraph(5); g AS(5); GQ(4, 6): Graph on 125 vertices sage: g.is_strongly_regular(parameters=True) (125, 28, 3, 7) sage: g=graphs.AhrensSzekeresGeneralizedQuadrangleGraph(5,dual=True); g AS(5)*; GQ(6, 4): Graph on 175 vertices sage: g.is_strongly_regular(parameters=True) (175, 30, 5, 5)
- sage.graphs.generators.classical_geometries.CossidentePenttilaGraph(q)¶
Cossidente-Penttila \(((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)\)-strongly regular graph
For each odd prime power \(q\), one can partition the points of the \(O_6^-(q)\)-generalized quadrangle \(GQ(q,q^2)\) into two parts, so that on any of them the induced subgraph of the point graph of the GQ has parameters as above [CP2005].
Directly following the construction in [CP2005] is not efficient, as one then needs to construct the dual \(GQ(q^2,q)\). Thus we describe here a more efficient approach that we came up with, following a suggestion by T.Penttila. Namely, this partition is invariant under the subgroup \(H=\Omega_3(q^2)<O_6^-(q)\). We build the appropriate \(H\), which leaves the form \(B(X,Y,Z)=XY+Z^2\) invariant, and pick up two orbits of \(H\) on the \(F_q\)-points. One them is \(B\)-isotropic, and we take the representative \((1:0:0)\). The other one corresponds to the points of \(PG(2,q^2)\) that have all the lines on them either missing the conic specified by \(B\), or intersecting the conic in two points. We take \((1:1:e)\) as the representative. It suffices to pick \(e\) so that \(e^2+1\) is not a square in \(F_{q^2}\). Indeed, The conic can be viewed as the union of \(\{(0:1:0)\}\) and \(\{(1:-t^2:t) | t \in F_{q^2}\}\). The coefficients of a generic line on \((1:1:e)\) are \([1:-1-eb:b]\), for \(-1\neq eb\). Thus, to make sure the intersection with the conic is always even, we need that the discriminant of \(1+(1+eb)t^2+tb=0\) never vanishes, and this is if and only if \(e^2+1\) is not a square. Further, we need to adjust \(B\), by multiplying it by appropriately chosen \(\nu\), so that \((1:1:e)\) becomes isotropic under the relative trace norm \(\nu B(X,Y,Z)+(\nu B(X,Y,Z))^q\). The latter is used then to define the graph.
INPUT:
q
– an odd prime power.
EXAMPLES:
For \(q=3\) one gets Sims-Gewirtz graph.
sage: G=graphs.CossidentePenttilaGraph(3) # optional - gap_packages (grape) sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape) (56, 10, 0, 2)
For \(q>3\) one gets new graphs.
sage: G=graphs.CossidentePenttilaGraph(5) # optional - gap_packages (grape) sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape) (378, 52, 1, 8)
- sage.graphs.generators.classical_geometries.HaemersGraph(q, hyperoval=None, hyperoval_matching=None, field=None, check_hyperoval=True)¶
Return the Haemers graph obtained from \(T_2^*(q)^*\)
Let \(q\) be a power of 2. In Sect. 8.A of [BL1984] one finds a construction of a strongly regular graph with parameters \((q^2(q+2),q^2+q-1,q-2,q)\) from the graph of \(T_2^*(q)^*\), constructed by
T2starGeneralizedQuadrangleGraph()
, by redefining adjacencies in the way specified by an arbitraryhyperoval_matching
of the points (i.e. partitioning into size two parts) ofhyperoval
defining \(T_2^*(q)^*\).While [BL1984] gives the construction in geometric terms, it can be formulated, and is implemented, in graph-theoretic ones, of re-adjusting the edges. Namely, \(G=T_2^*(q)^*\) has a partition into \(q+2\) independent sets \(I_k\) of size \(q^2\) each. Each vertex in \(I_j\) is adjacent to \(q\) vertices from \(I_k\). Each \(I_k\) is paired to some \(I_{k'}\), according to
hyperoval_matching
. One adds edges \((s,t)\) for \(s,t \in I_k\) whenever \(s\) and \(t\) are adjacent to some \(u \in I_{k'}\), and removes all the edges between \(I_k\) and \(I_{k'}\).INPUT:
q
– a power of twohyperoval_matching
– ifNone
(default), pair each \(i\)-th point ofhyperoval
with \((i+1)\)-th. Otherwise, specifies the pairing in the format \(((i_1,i'_1),(i_2,i'_2),...)\).hyperoval
– a hyperoval defining \(T_2^*(q)^*\). IfNone
(default), the classical hyperoval obtained from a conic is used. See the documentation ofT2starGeneralizedQuadrangleGraph()
, for more information.field
– an instance of a finite field of order \(q\), must be provided ifhyperoval
is provided.check_hyperoval
– (default:True
) ifTrue
, checkhyperoval
for correctness.
EXAMPLES:
using the built-in constructions:
sage: g=graphs.HaemersGraph(4); g Haemers(4): Graph on 96 vertices sage: g.is_strongly_regular(parameters=True) (96, 19, 2, 4)
supplying your own hyperoval_matching:
sage: g=graphs.HaemersGraph(4,hyperoval_matching=((0,5),(1,4),(2,3))); g Haemers(4): Graph on 96 vertices sage: g.is_strongly_regular(parameters=True) (96, 19, 2, 4)
- sage.graphs.generators.classical_geometries.NonisotropicOrthogonalPolarGraph(m, q, sign='+', perp=None)¶
Returns the Graph \(NO^{\epsilon,\perp}_{m}(q)\)
Let the vectorspace of dimension \(m\) over \(F_q\) be endowed with a nondegenerate quadratic form \(F\), of type
sign
for \(m\) even.\(m\) even: assume further that \(q=2\) or \(3\). Returns the graph of the points (in the underlying projective space) \(x\) satisfying \(F(x)=1\), with adjacency given by orthogonality w.r.t. \(F\). Parameter
perp
is ignored.\(m\) odd: if
perp
is notNone
, then we assume that \(q=5\) and return the graph of the points \(x\) satisfying \(F(x)=\pm 1\) ifsign="+"
, respectively \(F(x) \in \{2,3\}\) ifsign="-"
, with adjacency given by orthogonality w.r.t. \(F\) (cf. Sect 7.D of [BL1984]). Otherwise return the graph of nongenerate hyperplanes of typesign
, adjacent whenever the intersection is degenerate (cf. Sect. 7.C of [BL1984]). Note that for \(q=2\) one will get a complete graph.
For more information, see Sect. 9.9 of [BH2012] and [BL1984]. Note that the page of Andries Brouwer’s website uses different notation.
INPUT:
m
- integer, half the dimension of the underlying vectorspaceq
- a power of a prime number, the size of the underlying fieldsign
–"+"
(default) or"-"
.
EXAMPLES:
\(NO^-(4,2)\) is isomorphic to Petersen graph:
sage: g=graphs.NonisotropicOrthogonalPolarGraph(4,2,'-'); g NO^-(4, 2): Graph on 10 vertices sage: g.is_strongly_regular(parameters=True) (10, 3, 0, 1)
\(NO^-(6,2)\) and \(NO^+(6,2)\):
sage: g=graphs.NonisotropicOrthogonalPolarGraph(6,2,'-') sage: g.is_strongly_regular(parameters=True) (36, 15, 6, 6) sage: g=graphs.NonisotropicOrthogonalPolarGraph(6,2,'+'); g NO^+(6, 2): Graph on 28 vertices sage: g.is_strongly_regular(parameters=True) (28, 15, 6, 10)
\(NO^+(8,2)\):
sage: g=graphs.NonisotropicOrthogonalPolarGraph(8,2,'+') sage: g.is_strongly_regular(parameters=True) (120, 63, 30, 36)
Wilbrink’s graphs for \(q=5\):
sage: graphs.NonisotropicOrthogonalPolarGraph(5,5,perp=1).is_strongly_regular(parameters=True) # long time (325, 60, 15, 10) sage: graphs.NonisotropicOrthogonalPolarGraph(5,5,'-',perp=1).is_strongly_regular(parameters=True) # long time (300, 65, 10, 15)
Wilbrink’s graphs:
sage: g=graphs.NonisotropicOrthogonalPolarGraph(5,4,'+') sage: g.is_strongly_regular(parameters=True) (136, 75, 42, 40) sage: g=graphs.NonisotropicOrthogonalPolarGraph(5,4,'-') sage: g.is_strongly_regular(parameters=True) (120, 51, 18, 24) sage: g=graphs.NonisotropicOrthogonalPolarGraph(7,4,'+'); g # not tested (long time) NO^+(7, 4): Graph on 2080 vertices sage: g.is_strongly_regular(parameters=True) # not tested (long time) (2080, 1071, 558, 544)
- sage.graphs.generators.classical_geometries.NonisotropicUnitaryPolarGraph(m, q)¶
Returns the Graph \(NU(m,q)\).
Returns the graph on nonisotropic, with respect to a nondegenerate Hermitean form, points of the \((m-1)\)-dimensional projective space over \(F_q\), with points adjacent whenever they lie on a tangent (to the set of isotropic points) line. For more information, see Sect. 9.9 of [BH2012] and series C14 in [Hub1975].
INPUT:
m,q
(integers) – \(q\) must be a prime power.
EXAMPLES:
sage: g=graphs.NonisotropicUnitaryPolarGraph(5,2); g NU(5, 2): Graph on 176 vertices sage: g.is_strongly_regular(parameters=True) (176, 135, 102, 108)
- sage.graphs.generators.classical_geometries.Nowhere0WordsTwoWeightCodeGraph(q, hyperoval=None, field=None, check_hyperoval=True)¶
Return the subgraph of nowhere 0 words from two-weight code of projective plane hyperoval.
Let \(q=2^k\) and \(\Pi=PG(2,q)\). Fix a hyperoval \(O \subset \Pi\). Let \(V=F_q^3\) and \(C\) the two-weight 3-dimensional linear code over \(F_q\) with words \(c(v)\) obtained from \(v\in V\) by computing
\[c(v)=(\langle v,o_1 \rangle,...,\langle v,o_{q+2} \rangle), o_j \in O.\]\(C\) contains \(q(q-1)^2/2\) words without 0 entries. The subgraph of the strongly regular graph of \(C\) induced on the latter words is also strongly regular, assuming \(q>4\). This is a construction due to A.E.Brouwer [Bro2016], and leads to graphs with parameters also given by a construction in [HHL2009]. According to [Bro2016], these two constructions are likely to produce isomorphic graphs.
INPUT:
q
– a power of twohyperoval
– a hyperoval (i.e. a complete 2-arc; a set of points in the plane meeting every line in 0 or 2 points) in \(PG(2,q)\) over the fieldfield
. Each point ofhyperoval
must be a length 3 vector overfield
with 1st non-0 coordinate equal to 1. By default,hyperoval
andfield
are not specified, and constructed on the fly. In particular,hyperoval
we build is the classical one, i.e. a conic with the point of intersection of its tangent lines.field
– an instance of a finite field of order \(q\), must be provided ifhyperoval
is provided.check_hyperoval
– (default:True
) ifTrue
, checkhyperoval
for correctness.
See also
EXAMPLES:
using the built-in construction:
sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(8); g Nowhere0WordsTwoWeightCodeGraph(8): Graph on 196 vertices sage: g.is_strongly_regular(parameters=True) (196, 60, 14, 20) sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(16) # not tested (long time) sage: g.is_strongly_regular(parameters=True) # not tested (long time) (1800, 728, 268, 312)
supplying your own hyperoval:
sage: F=GF(8) sage: O=[vector(F,(0,0,1)),vector(F,(0,1,0))]+[vector(F, (1,x^2,x)) for x in F] sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(8,hyperoval=O,field=F); g Nowhere0WordsTwoWeightCodeGraph(8): Graph on 196 vertices sage: g.is_strongly_regular(parameters=True) (196, 60, 14, 20)
- sage.graphs.generators.classical_geometries.OrthogonalDualPolarGraph(e, d, q)¶
Return dual polar graph on \(GO^e(n,q)\) of diameter \(d\). The value of \(n\) is determined by \(d\) and \(e\).
The graph is distance-regular with classical parameters \((d, q, 0, q^e)\).
INPUT:
e
– integer; type of the orthogonal polar space to consider; must be \(-1, 0\) or \(1\).d
– integer; diameter of the graphq
– integer; prime power; order of the finite field over which to build the polar space
EXAMPLES:
sage: G = graphs.OrthogonalDualPolarGraph(1,3,2) sage: G.is_distance_regular(True) ([7, 6, 4, None], [None, 1, 3, 7]) sage: G = graphs.OrthogonalDualPolarGraph(0,3,3) sage: G.is_distance_regular(True) ([39, 36, 27, None], [None, 1, 4, 13]) sage: G.order() 1120
REFERENCES:
See [BCN1989] pp. 274-279 or [VDKT2016] p. 22.
- sage.graphs.generators.classical_geometries.OrthogonalPolarGraph(m, q, sign='+')¶
Returns the Orthogonal Polar Graph \(O^{\epsilon}(m,q)\).
For more information on Orthogonal Polar graphs, see the page of Andries Brouwer’s website.
INPUT:
m,q
(integers) – \(q\) must be a prime power.sign
–"+"
or"-"
if \(m\) is even,"+"
(default) otherwise.
EXAMPLES:
sage: G = graphs.OrthogonalPolarGraph(6,3,"+"); G Orthogonal Polar Graph O^+(6, 3): Graph on 130 vertices sage: G.is_strongly_regular(parameters=True) (130, 48, 20, 16) sage: G = graphs.OrthogonalPolarGraph(6,3,"-"); G Orthogonal Polar Graph O^-(6, 3): Graph on 112 vertices sage: G.is_strongly_regular(parameters=True) (112, 30, 2, 10) sage: G = graphs.OrthogonalPolarGraph(5,3); G Orthogonal Polar Graph O(5, 3): Graph on 40 vertices sage: G.is_strongly_regular(parameters=True) (40, 12, 2, 4) sage: G = graphs.OrthogonalPolarGraph(8,2,"+"); G Orthogonal Polar Graph O^+(8, 2): Graph on 135 vertices sage: G.is_strongly_regular(parameters=True) (135, 70, 37, 35) sage: G = graphs.OrthogonalPolarGraph(8,2,"-"); G Orthogonal Polar Graph O^-(8, 2): Graph on 119 vertices sage: G.is_strongly_regular(parameters=True) (119, 54, 21, 27)
- sage.graphs.generators.classical_geometries.SymplecticDualPolarGraph(m, q)¶
Returns the Symplectic Dual Polar Graph \(DSp(m,q)\).
For more information on Symplectic Dual Polar graphs, see [BCN1989] and Sect. 2.3.1 of [Coh1981].
INPUT:
m,q
(integers) – \(q\) must be a prime power, and \(m\) must be even.
EXAMPLES:
sage: G = graphs.SymplecticDualPolarGraph(6,3); G # not tested (long time) Symplectic Dual Polar Graph DSp(6, 3): Graph on 1120 vertices sage: G.is_distance_regular(parameters=True) # not tested (long time) ([39, 36, 27, None], [None, 1, 4, 13])
- sage.graphs.generators.classical_geometries.SymplecticPolarGraph(d, q, algorithm=None)¶
Returns the Symplectic Polar Graph \(Sp(d,q)\).
The Symplectic Polar Graph \(Sp(d,q)\) is built from a projective space of dimension \(d-1\) over a field \(F_q\), and a symplectic form \(f\). Two vertices \(u,v\) are made adjacent if \(f(u,v)=0\).
See the page on symplectic graphs on Andries Brouwer’s website.
INPUT:
d,q
(integers) – note that only even values of \(d\) are accepted by the function.algorithm
– if set to ‘gap’ then the computation is carried via GAP library interface, computing totally singular subspaces, which is faster for \(q>3\). Otherwise it is done directly.
EXAMPLES:
Computation of the spectrum of \(Sp(6,2)\):
sage: g = graphs.SymplecticPolarGraph(6,2) sage: g.is_strongly_regular(parameters=True) (63, 30, 13, 15) sage: set(g.spectrum()) == {-5, 3, 30} True
The parameters of \(Sp(4,q)\) are the same as of \(O(5,q)\), but they are not isomorphic if \(q\) is odd:
sage: G = graphs.SymplecticPolarGraph(4,3) sage: G.is_strongly_regular(parameters=True) (40, 12, 2, 4) sage: O=graphs.OrthogonalPolarGraph(5,3) sage: O.is_strongly_regular(parameters=True) (40, 12, 2, 4) sage: O.is_isomorphic(G) False sage: graphs.SymplecticPolarGraph(6,4,algorithm="gap").is_strongly_regular(parameters=True) # not tested (long time) (1365, 340, 83, 85)
- sage.graphs.generators.classical_geometries.T2starGeneralizedQuadrangleGraph(q, dual=False, hyperoval=None, field=None, check_hyperoval=True)¶
Return the collinearity graph of the generalized quadrangle \(T_2^*(q)\), or of its dual
Let \(q=2^k\) and \(\Theta=PG(3,q)\). \(T_2^*(q)\) is a generalized quadrangle (Wikipedia article Generalized_quadrangle) of order \((q-1,q+1)\), see 3.1.3 in [PT2009]. Fix a plane \(\Pi \subset \Theta\) and a hyperoval \(O \subset \Pi\). The points of \(T_2^*(q):=T_2^*(O)\) are the points of \(\Theta\) outside \(\Pi\), and the lines are the lines of \(\Theta\) outside \(\Pi\) that meet \(\Pi\) in a point of \(O\).
INPUT:
q
– a power of twodual
– ifFalse
(default), return the graph of \(T_2^*(O)\). Otherwise return the graph of the dual \(T_2^*(O)\).hyperoval
– a hyperoval (i.e. a complete 2-arc; a set of points in the plane meeting every line in 0 or 2 points) in the plane of points with 0th coordinate 0 in \(PG(3,q)\) over the fieldfield
. Each point ofhyperoval
must be a length 4 vector overfield
with 1st non-0 coordinate equal to 1. By default,hyperoval
andfield
are not specified, and constructed on the fly. In particular,hyperoval
we build is the classical one, i.e. a conic with the point of intersection of its tangent lines.field
– an instance of a finite field of order \(q\), must be provided ifhyperoval
is provided.check_hyperoval
– (default:True
) ifTrue
, checkhyperoval
for correctness.
EXAMPLES:
using the built-in construction:
sage: g=graphs.T2starGeneralizedQuadrangleGraph(4); g T2*(O,4); GQ(3, 5): Graph on 64 vertices sage: g.is_strongly_regular(parameters=True) (64, 18, 2, 6) sage: g=graphs.T2starGeneralizedQuadrangleGraph(4,dual=True); g T2*(O,4)*; GQ(5, 3): Graph on 96 vertices sage: g.is_strongly_regular(parameters=True) (96, 20, 4, 4)
supplying your own hyperoval:
sage: F=GF(4,'b') sage: O=[vector(F,(0,0,0,1)),vector(F,(0,0,1,0))]+[vector(F, (0,1,x^2,x)) for x in F] sage: g=graphs.T2starGeneralizedQuadrangleGraph(4, hyperoval=O, field=F); g T2*(O,4); GQ(3, 5): Graph on 64 vertices sage: g.is_strongly_regular(parameters=True) (64, 18, 2, 6)
- sage.graphs.generators.classical_geometries.TaylorTwographDescendantSRG(q, clique_partition=None)¶
constructing the descendant graph of the Taylor’s two-graph for \(U_3(q)\), \(q\) odd
This is a strongly regular graph with parameters \((v,k,\lambda,\mu)=(q^3, (q^2+1)(q-1)/2, (q-1)^3/4-1, (q^2+1)(q-1)/4)\) obtained as a two-graph descendant of the
Taylor's two-graph
\(T\). This graph admits a partition into cliques of size \(q\), which are useful inTaylorTwographSRG()
, a strongly regular graph on \(q^3+1\) vertices in the Seidel switching class of \(T\), for which we need \((q^2+1)/2\) cliques. The cliques are the \(q^2\) lines on \(v_0\) of the projective plane containing the unital for \(U_3(q)\), and intersecting the unital (i.e. the vertices of the graph and the point we remove) in \(q+1\) points. This is all taken from §7E of [BL1984].INPUT:
q
– a power of an odd prime numberclique_partition
– ifTrue
, return \(q^2-1\) cliques of size \(q\) with empty pairwise intersection. (Removing all of them leaves a clique, too), and the point removed from the unital.
EXAMPLES:
sage: g=graphs.TaylorTwographDescendantSRG(3); g Taylor two-graph descendant SRG: Graph on 27 vertices sage: g.is_strongly_regular(parameters=True) (27, 10, 1, 5) sage: from sage.combinat.designs.twographs import taylor_twograph sage: T = taylor_twograph(3) # long time sage: g.is_isomorphic(T.descendant(T.ground_set()[1])) # long time True sage: g=graphs.TaylorTwographDescendantSRG(5) # not tested (long time) sage: g.is_strongly_regular(parameters=True) # not tested (long time) (125, 52, 15, 26)
- sage.graphs.generators.classical_geometries.TaylorTwographSRG(q)¶
constructing a strongly regular graph from the Taylor’s two-graph for \(U_3(q)\), \(q\) odd
This is a strongly regular graph with parameters \((v,k,\lambda,\mu)=(q^3+1, q(q^2+1)/2, (q^2+3)(q-1)/4, (q^2+1)(q+1)/4)\) in the Seidel switching class of
Taylor two-graph
. Details are in §7E of [BL1984].INPUT:
q
– a power of an odd prime number
See also
EXAMPLES:
sage: t=graphs.TaylorTwographSRG(3); t Taylor two-graph SRG: Graph on 28 vertices sage: t.is_strongly_regular(parameters=True) (28, 15, 6, 10)
- sage.graphs.generators.classical_geometries.UnitaryDualPolarGraph(m, q)¶
Returns the Dual Unitary Polar Graph \(U(m,q)\).
For more information on Unitary Dual Polar graphs, see [BCN1989] and Sect. 2.3.1 of [Coh1981].
INPUT:
m,q
(integers) – \(q\) must be a prime power.
EXAMPLES:
The point graph of a generalized quadrangle (see Wikipedia article Generalized_quadrangle, [PT2009]) of order (8,4):
sage: G = graphs.UnitaryDualPolarGraph(5,2); G # long time Unitary Dual Polar Graph DU(5, 2); GQ(8, 4): Graph on 297 vertices sage: G.is_strongly_regular(parameters=True) # long time (297, 40, 7, 5)
Another way to get the generalized quadrangle of order (2,4):
sage: G = graphs.UnitaryDualPolarGraph(4,2); G Unitary Dual Polar Graph DU(4, 2); GQ(2, 4): Graph on 27 vertices sage: G.is_isomorphic(graphs.OrthogonalPolarGraph(6,2,'-')) True
A bigger graph:
sage: G = graphs.UnitaryDualPolarGraph(6,2); G # not tested (long time) Unitary Dual Polar Graph DU(6, 2): Graph on 891 vertices sage: G.is_distance_regular(parameters=True) # not tested (long time) ([42, 40, 32, None], [None, 1, 5, 21])
- sage.graphs.generators.classical_geometries.UnitaryPolarGraph(m, q, algorithm='gap')¶
Returns the Unitary Polar Graph \(U(m,q)\).
For more information on Unitary Polar graphs, see the page of Andries Brouwer’s website.
INPUT:
m,q
(integers) – \(q\) must be a prime power.algorithm
– if set to ‘gap’ then the computation is carried via GAP library interface, computing totally singular subspaces, which is faster for large examples (especially with \(q>2\)). Otherwise it is done directly.
EXAMPLES:
sage: G = graphs.UnitaryPolarGraph(4,2); G Unitary Polar Graph U(4, 2); GQ(4, 2): Graph on 45 vertices sage: G.is_strongly_regular(parameters=True) (45, 12, 3, 3) sage: graphs.UnitaryPolarGraph(5,2).is_strongly_regular(parameters=True) (165, 36, 3, 9) sage: graphs.UnitaryPolarGraph(6,2) # not tested (long time) Unitary Polar Graph U(6, 2): Graph on 693 vertices