Designs and other combinatorial structures: designs
and arrays, single-change circular covering designs, sequential covering designs,
orthogonal arrays, properly separated permutations, and Latin squares; have
invented a board game called squares. Designs are used worldwide
by agricultural scientists when testing new fertilizers, pharmaceutical companies
in testing new drugs, and sports organizations for arranging game schedules,
enumeration, coloring, graph sequences and digraphs, magic labelings,
injections, neighborhood properties. Graphs are used by chemists,
the business community, telecommunications companies, etc.,
to model chemical structures, small economies, telephone networks, etc.
Any advance in the theory of graphs has potential benefits for these people.
Combinatorial interpretations of polynomials:
Bessel polynomials and derivatives, Lommel polynomials and derivatives, multivariate
matchings polynomials of graphs, m-path cover polynomials of graphs,
vertex/matching-partition function of graphs.
Rhombic tilings of (n,k)-ovals, (n,k,λ)-cyclic difference sets, and related topics,
with A.Schoen. Discrete Mathematics 313 (2013), pp. 129-154.
Zeons, permanents, the Johnson scheme, and generalised derangements,
with P.Feinsilver. International Journal of Combinatorics, (2011), v.2011, Article ID 539030, 29 pages.
On k-minimum and m-minimum Edge Magic Injections of Graphs.
with J.Trono. Discrete Mathematics.(2010) v.310 (no.1) pp.56–69
Multivariate Matching Polynomials of Cyclically Labelled Graphs,
with P.Feinsilver. Discrete Mathematics. (2009) v.309 pp.3205–3218