239 research outputs found
Every generalized quadrangle of order 5 having a regular point is symplectic
For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof
Identifying codes in vertex-transitive graphs and strongly regular graphs
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 2ln(vertical bar V vertical bar) + 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 vertical bar V vertical bar(alpha) with alpha is an element of{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
Point regular groups of automorphisms of generalised quadrangles
We study the point regular groups of automorphisms of some of the known
generalised quadrangles. In particular we determine all point regular groups of
automorphisms of the thick classical generalised quadrangles. We also construct
point regular groups of automorphisms of the generalised quadrangle of order
obtained by Payne derivation from the classical symplectic
quadrangle . For with we obtain at least two
nonisomorphic groups when and at least three nonisomorphic groups
when or . Our groups include nonabelian 2-groups, groups of exponent 9
and nonspecial -groups. We also enumerate all point regular groups of
automorphisms of some small generalised quadrangles.Comment: some minor changes (including to title) after referee's comment
On hyperovals of polar spaces
We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)
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