125,589 research outputs found
Distance-regular Cayley graphs over (pseudo-) semi-dihedral groups
Distance-regular graphs are a class of regualr graphs with pretty
combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the
problem of charaterizing distance-regular Cayley graphs, which can be viewed as
a natural extension of the problem of characterizing strongly-regular Cayley
graphs (or equivalently, regular partial difference sets). In this paper, we
provide a partial characterization for distance-regular Cayley graphs over
semi-dihedral groups and pseudo-semi-dihedral groups, both of which are
-groups with a cyclic subgroup of index .Comment: 21 page
Switching with more than two colours
AbstractThe operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m>2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case.This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive.In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random m-coloured complete graph) where the group of switching-automorphisms is highly transitive.Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds
Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals
The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, in which every node has a degree from an interval not too far away from a given . In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, we provide the first non-trivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given near-regular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly log-concave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\"uller-Hannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of near-regular degree intervals of size at most one
Equiangular lines via matrix projection
In 1973, Lemmens and Seidel posed the problem of determining the maximum
number of equiangular lines in with angle and
gave a partial answer in the regime . At the other
extreme where is at least exponential in , recent breakthroughs
have led to an almost complete resolution of this problem. In this paper, we
introduce a new method for obtaining upper bounds which unifies and improves
upon previous approaches, thereby bridging the gap between the aforementioned
regimes, as well as significantly extending or improving all previously known
bounds when . Our method is based on orthogonal
projection of matrices with respect to the Frobenius inner product and it also
yields the first extension of the Alon-Boppana theorem to dense graphs, with
equality for strongly regular graphs corresponding to
equiangular lines in . Applications of our method in the complex
setting will be discussed as well.Comment: 39 pages, LaTeX; added new and improved results, improved
presentatio
Symmetry adapted Assur decompositions
Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited the orbit matrix as a symmetry
adapted form of the rigidity matrix. This paper explores how the decomposition
and analysis of symmetric frameworks and their symmetric motions can be
supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure
On strictly Deza graphs with parameters (n,k,k-1,a)
A nonempty -regular graph on vertices is called a Deza graph
if there exist constants and such that any pair of
distinct vertices of has precisely either or common
neighbours. The quantities , , , and are called the parameters of
and are written as the quadruple . If a Deza graph has
diameter 2 and is not strongly regular, then it is called a strictly Deza
graph. In the paper we investigate strictly Deza graphs with parameters , where its quantities satisfy the conditions and
.Comment: Any comments or suggestions are very welcom
The strongly regular (45,12,3,3) graphs
Using two backtrack algorithms based on dierent techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters v = 45, k = 12, λ = μ = 3. It turns out that there are 78 such graphs, having automorphism groups with sizes ranging from 1 to 51840
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