We give a general construction leading to different non-isomorphic families of connected q-regular semisymmetric graphs of order 2q (n+1) embedded in , for a prime power q=p (h) , using the linear representation of a particular point set of size q contained in a hyperplane of . We show that, when is a normal rational curve with one point removed, the graphs are isomorphic to the graphs constructed for q=p (h) in Lazebnik and Viglione (J. Graph Theory 41, 249-258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897-902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For qa parts per thousand yenn+3 or q=p=n+2, na parts per thousand yen2, we obtain their full automorphism group from our construction by showing that, for an arc , every automorphism of is induced by a collineation of the ambient space . We also give some other examples of semisymmetric graphs for which not every automorphism is induced by a collineation of their ambient space
The linear representation T∗n(K) of a point set K in a hyperplane of PG(n+1,q) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations T∗n(K) and T∗n(K′), under a few conditions on K and K′. First, we prove that an isomorphism between T∗n(K) and T∗n(K′) is induced by an isomorphism between the two linear representations T∗n(K) and T∗n(K′) of their closures K and K′. This allows us to focus on the automorphism group of a linear representation T∗n(S) of a subgeometry S≅PG(n,q) embedded in a hyperplane of the projective space PG(n+1,qt). To this end we introduce a geometry X(n,t,q) and determine its automorphism group. The geometry X(n,t,q) is a straightforward generalization of Hn+2q which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of X(n,t,q) as a coset geometry we extend this result and prove that X(n,t,q) and T∗n(S) are isomorphic. Finally, we compare the full automorphism group of T∗n(S) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space
It is known that the classical unital arising from the Hermitian curve in PG(2, 9) does not have a 2-coloring without monochromatic lines. Here we show that for q ≥ 4, the Hermitian curve in PG(2, q 2 ) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital.
Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set A of (n − 1)-spaces in PG(3n − 1, q) such that any three span the whole space. Pseudo-arcs of size q n + 1 are called pseudo-ovals, while pseudo-arcs of size q n + 2 are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in PG(2, q n ).We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in PG(3n − 1, q), where q is even and n is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes.
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V |) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V | α with α ∈ { 1 4 , 1 3 , 2 5 }. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.
A linear representation T * n (K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1, q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of T * n (K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations T * n (K) and T * n (K ) are isomorphic if and only if the point sets K and K are PΓL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of T * n (K) that are induced by collineations of PG(n + 1, q).
For n ≥ 9, we construct maximal partial line spreads for non-singular quadrics of P G(n, q) for every size between approximately (cn + d)(q n−3 + q n−5 ) log 2q and q n−2 , for some small constants c and d. These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gács and Szőnyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles W 3 (q) and Q(4, q) by Pepe, Rößing and Storme.
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