134 research outputs found
Tutte polynomial of a small-world farey graph
In this paper, we find recursive formulas for the Tutte polynomial of a
family of small-world networks: Farey graphs, which are modular and have an
exponential degree hierarchy. Then, making use of these formulas, we determine
the number of spanning trees, as well as the number of connected spanning
subgraphs. Furthermore, we also derive exact expressions for the chromatic
polynomial and the reliability polynomial of these graphs.Comment: 6 page
Order Quasisymmetric Functions Distinguish Rooted Trees
Richard P. Stanley conjectured that finite trees can be distinguished by
their chromatic symmetric functions. In this paper, we prove an analogous
statement for posets: Finite rooted trees can be distinguished by their order
quasisymmetric functions.Comment: 16 pages, 5 figures, referees' suggestions incorporate
Overlapping Community Structure in Co-authorship Networks: a Case Study
Community structure is one of the key properties of real-world complex
networks. It plays a crucial role in their behaviors and topology. While an
important work has been done on the issue of community detection, very little
attention has been devoted to the analysis of the community structure. In this
paper, we present an extensive investigation of the overlapping community
network deduced from a large-scale co-authorship network. The nodes of the
overlapping community network represent the functional communities of the
co-authorship network, and the links account for the fact that communities
share some nodes in the co-authorship network. The comparative evaluation of
the topological properties of these two networks shows that they share similar
topological properties. These results are very interesting. Indeed, the network
of communities seems to be a good representative of the original co-authorship
network. With its smaller size, it may be more practical in order to realize
various analyses that cannot be performed easily in large-scale real-world
networks.Comment: 2014 7th International Conference on u- and e- Service, Science and
Technolog
A note on the convexity number for complementary prisms
In the geodetic convexity, a set of vertices of a graph is
if all vertices belonging to any shortest path between two
vertices of lie in . The cardinality of a maximum proper convex
set of is the of . The
of a graph arises from the
disjoint union of the graph and by adding the edges of a
perfect matching between the corresponding vertices of and .
In this work, we we prove that the decision problem related to the convexity
number is NP-complete even restricted to complementary prisms, we determine
when is disconnected or is a cograph, and we
present a lower bound when .Comment: 10 pages, 2 figure
The zero forcing polynomial of a graph
Zero forcing is an iterative graph coloring process, where given a set of
initially colored vertices, a colored vertex with a single uncolored neighbor
causes that neighbor to become colored. A zero forcing set is a set of
initially colored vertices which causes the entire graph to eventually become
colored. In this paper, we study the counting problem associated with zero
forcing. We introduce the zero forcing polynomial of a graph of order
as the polynomial , where is
the number of zero forcing sets of of size . We characterize the
extremal coefficients of , derive closed form expressions for
the zero forcing polynomials of several families of graphs, and explore various
structural properties of , including multiplicativity,
unimodality, and uniqueness.Comment: 23 page
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