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The Orchard crossing number of an abstract graph
We introduce the Orchard crossing number, which is defined in a similar way
to the well-known rectilinear crossing number. We compute the Orchard crossing
number for some simple families of graphs. We also prove some properties of
this crossing number.
Moreover, we define a variant of this crossing number which is tightly
connected to the rectilinear crossing number, and compute it for some simple
families of graphs.Comment: 17 pages, 10 figures. Totally revised, new material added. Submitte
Integration and measures on the space of countable labelled graphs
In this paper we develop a rigorous foundation for the study of integration
and measures on the space of all graphs defined on a countable
labelled vertex set . We first study several interrelated -algebras
and a large family of probability measures on graph space. We then focus on a
"dyadic" Hamming distance function , which was
very useful in the study of differentiation on . The function
is shown to be a Haar measure-preserving
bijection from the subset of infinite graphs to the circle (with the
Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a
consequence, we establish a "change of variables" formula that enables the
transfer of the Riemann-Lebesgue theory on to graph space
. This also complements previous work in which a theory of
Newton-Leibnitz differentiation was transferred from the real line to
for countable . Finally, we identify the Pontryagin dual of
, and characterize the positive definite functions on
.Comment: 15 pages, LaTe
Ideal Graph of a Graph
In this paper, we introduce ideal graph of a graph and study some of its properties.
We characterize connectedness, isomorphism of graphs and coloring property of a
graph using ideal graph. Also, we give an upper bound for chromatic number of a graph
Graph properties of graph associahedra
A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction
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