1,025 research outputs found
Farey Graphs as Models for Complex Networks
Farey sequences of irreducible fractions between 0 and 1 can be related to
graph constructions known as Farey graphs. These graphs were first introduced
by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they
have many interesting properties: they are minimally 3-colorable, uniquely
Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a
simple generation method for a Farey graph family, and we study analytically
relevant topological properties: order, size, degree distribution and
correlation, clustering, transitivity, diameter and average distance. We show
that the graphs are a good model for networks associated with some complex
systems.Comment: Definitive version published in Theoretical Computer Scienc
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
Farey graphs as models for complex networks
Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs.
These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect.
In this paper we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.Peer Reviewe
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