13,132 research outputs found
Prime Graphs and Exponential Composition of Species
In this paper, we enumerate prime graphs with respect to the Cartesian
multiplication of graphs. We use the unique factorization of a connected graph
into the product of prime graphs given by Sabidussi to find explicit formulas
for labeled and unlabeled prime graphs. In the case of species, we construct
the exponential composition of species based on the arithmetic product of
species of Maia and M\'endez and the quotient species, and express the species
of connected graphs as the exponential composition of the species of prime
graphs.Comment: 30 pages, 7 figures, 1 tabl
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Metastability and anomalous fixation in evolutionary games on scale-free networks
We study the influence of complex graphs on the metastability and fixation
properties of a set of evolutionary processes. In the framework of evolutionary
game theory, where the fitness and selection are frequency-dependent and vary
with the population composition, we analyze the dynamics of snowdrift games
(characterized by a metastable coexistence state) on scale-free networks. Using
an effective diffusion theory in the weak selection limit, we demonstrate how
the scale-free structure affects the system's metastable state and leads to
anomalous fixation. In particular, we analytically and numerically show that
the probability and mean time of fixation are characterized by stretched
exponential behaviors with exponents depending on the network's degree
distribution.Comment: 5 pages, 4 figures, to appear in Physical Review Letter
Some results on more flexible versions of Graph Motif
The problems studied in this paper originate from Graph Motif, a problem
introduced in 2006 in the context of biological networks. Informally speaking,
it consists in deciding if a multiset of colors occurs in a connected subgraph
of a vertex-colored graph. Due to the high rate of noise in the biological
data, more flexible definitions of the problem have been outlined. We present
in this paper two inapproximability results for two different optimization
variants of Graph Motif: one where the size of the solution is maximized, the
other when the number of substitutions of colors to obtain the motif from the
solution is minimized. We also study a decision version of Graph Motif where
the connectivity constraint is replaced by the well known notion of graph
modularity. While the problem remains NP-complete, it allows algorithms in FPT
for biologically relevant parameterizations
{\Gamma}-species, quotients, and graph enumeration
The theory of {\Gamma}-species is developed to allow species-theoretic study
of quotient structures in a categorically rigorous fashion. This new approach
is then applied to two graph-enumeration problems which were previously
unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio
Statistics on Graphs, Exponential Formula and Combinatorial Physics
The concern of this paper is a famous combinatorial formula known under the
name "exponential formula". It occurs quite naturally in many contexts
(physics, mathematics, computer science). Roughly speaking, it expresses that
the exponential generating function of a whole structure is equal to the
exponential of those of connected substructures. Keeping this descriptive
statement as a guideline, we develop a general framework to handle many
different situations in which the exponential formula can be applied
Arithmetical Semirings
We study the number of connected graphs with vertices that cannot be
written as the cartesian product of two graphs with fewer vertices. We give an
upper bound which implies that for large almost all graphs are both
connected and cartesian prime. For graphs with an even number of vertices, a
full asymptotic expansion is obtained. Our method, inspired by Knopfmacher's
theory of arithmetical semigroups, is based on reduction to Wright's asymptotic
expansion for the number of connected graphs with vertices.Comment: 18 pages, enhanced exposition, minor correction
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