530 research outputs found
On highly regular strongly regular graphs
In this paper we unify several existing regularity conditions for graphs,
including strong regularity, -isoregularity, and the -vertex condition.
We develop an algebraic composition/decomposition theory of regularity
conditions. Using our theoretical results we show that a family of non rank 3
graphs known to satisfy the -vertex condition fulfills an even stronger
condition, -regularity (the notion is defined in the text). Derived from
this family we obtain a new infinite family of non rank strongly regular
graphs satisfying the -vertex condition. This strengthens and generalizes
previous results by Reichard.Comment: 29 page
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
Direct constructions of hyperplanes of dual polar spaces arising from embeddings
Let e be one of the following full projective embeddings of a finite dual polar space Delta of rank n >= 2: (i) The Grassmann-embedding of the symplectic dual polar space Delta congruent to DW(2n 1,q); (ii) the Grassmann-embedding of the Hermitian dual polar space Delta congruent to DH(2n-1, q(2)); (iii) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(-)(2n+ 1, q). Let H-e denote the set of all hyperplanes of Delta arising from the embedding e. We give a method for constructing the hyperplanes of H-e without implementing the embedding e and discuss (possible) applications of the given construction
On a conjecture of Brouwer involving the connectivity of strongly regular graphs
In this paper, we study a conjecture of Andries E. Brouwer from 1996
regarding the minimum number of vertices of a strongly regular graph whose
removal disconnects the graph into non-singleton components.
We show that strongly regular graphs constructed from copolar spaces and from
the more general spaces called -spaces are counterexamples to Brouwer's
Conjecture. Using J.I. Hall's characterization of finite reduced copolar
spaces, we find that the triangular graphs , the symplectic graphs
over the field (for any prime power), and the
strongly regular graphs constructed from the hyperbolic quadrics
and from the elliptic quadrics over the field ,
respectively, are counterexamples to Brouwer's Conjecture. For each of these
graphs, we determine precisely the minimum number of vertices whose removal
disconnects the graph into non-singleton components. While we are not aware of
an analogue of Hall's characterization theorem for -spaces, we show
that complements of the point graphs of certain finite generalized quadrangles
are point graphs of -spaces and thus, yield other counterexamples to
Brouwer's Conjecture.
We prove that Brouwer's Conjecture is true for many families of strongly
regular graphs including the conference graphs, the generalized quadrangles
graphs, the lattice graphs, the Latin square graphs, the strongly
regular graphs with smallest eigenvalue -2 (except the triangular graphs) and
the primitive strongly regular graphs with at most 30 vertices except for few
cases.
We leave as an open problem determining the best general lower bound for the
minimum size of a disconnecting set of vertices of a strongly regular graph,
whose removal disconnects the graph into non-singleton components.Comment: 25 pages, 1 table; accepted to JCTA; revised version contains a new
section on copolar and Delta space
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