40,590 research outputs found
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
Softening Transitions with Quenched 2D Gravity
We perform extensive Monte Carlo simulations of the 10-state Potts model on
quenched two-dimensional gravity graphs to study the effect of
quenched connectivity disorder on the phase transition, which is strongly first
order on regular lattices. The numerical data provides strong evidence that,
due to the quenched randomness, the discontinuous first-order phase transition
of the pure model is softened to a continuous transition.Comment: 3 pages, LaTeX + 1 postscript figure. Talk presented at
LATTICE96(other models). See also
http://www.cond-mat.physik.uni-mainz.de/~janke/doc/home_janke.htm
Softening of First-Order Phase Transition on Quenched Random Gravity Graphs
We perform extensive Monte Carlo simulations of the 10-state Potts model on
quenched two-dimensional gravity graphs to study the effect of
quenched coordination number randomness on the nature of the phase transition,
which is strongly first order on regular lattices. The numerical data provides
strong evidence that, due to the quenched randomness, the discontinuous
first-order phase transition of the pure model is softened to a continuous
transition, representing presumably a new universality class. This result is in
striking contrast to a recent Monte Carlo study of the 8-state Potts model on
two-dimensional Poissonian random lattices of Voronoi/Delaunay type, where the
phase transition clearly stayed of first order, but is in qualitative agreement
with results for quenched bond randomness on regular lattices. A precedent for
such softening with connectivity disorder is known: in the 10-state Potts model
on annealed Phi3 gravity graphs a continuous transition is also observed.Comment: Latex + 5 postscript figures, 10 pages of text, figures appende
Certain properties of the enhanced power graph associated with a finite group
The enhanced power graph of a finite group , denoted by
, is the simple undirected graph whose vertex set is and
two distinct vertices are adjacent if for
some . In this article, we determine all finite groups such that the
minimum degree and the vertex connectivity of are equal.
Also, we classify all groups whose (proper) enhanced power graphs are strongly
regular. Further, the vertex connectivity of the enhanced power graphs
associated to some nilpotent groups is obtained. Finally, we obtain a lower
bound and an upper bound for the Wiener index of , where
is a nilpotent group. The finite nilpotent groups attaining these bounds are
also characterized.Comment: arXiv admin note: text overlap with arXiv:2207.0464
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