63 research outputs found
Structural Differentiation of Graphs Using Hosoya-Based Indices
In this paper, we introduce the Hosoya-Spectral indices and the Hosoya information content of a graph. The first measure combines structural information captured by partial Hosoya polynomials and graph spectra. The latter is a graph entropy measure which is based on blocks consisting of vertices with the same partial Hosoya polynomial. We evaluate the discrimination power of these quantities by interpreting numerical results
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
A faster hafnian formula for complex matrices and its benchmarking on a supercomputer
We introduce new and simple algorithms for the calculation of the number of
perfect matchings of complex weighted, undirected graphs with and without
loops. Our compact formulas for the hafnian and loop hafnian of
complex matrices run in time, are embarrassingly
parallelizable and, to the best of our knowledge, are the fastest exact
algorithms to compute these quantities. Despite our highly optimized algorithm,
numerical benchmarks on the Titan supercomputer with matrices up to size indicate that one would require the 288000 CPUs of this machine for
about a month and a half to compute the hafnian of a matrix.Comment: 11 pages, 7 figures. The source code of the library is available at
https://github.com/XanaduAI/hafnian . Accepted for publication in Journal of
Experimental Algorithmic
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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