1,793 research outputs found
Separation dimension of bounded degree graphs
The 'separation dimension' of a graph is the smallest natural number
for which the vertices of can be embedded in such that any
pair of disjoint edges in can be separated by a hyperplane normal to one of
the axes. Equivalently, it is the smallest possible cardinality of a family
of total orders of the vertices of such that for any two
disjoint edges of , there exists at least one total order in
in which all the vertices in one edge precede those in the other. In general,
the maximum separation dimension of a graph on vertices is . In this article, we focus on bounded degree graphs and show that the
separation dimension of a graph with maximum degree is at most
. We also demonstrate that the above bound is nearly
tight by showing that, for every , almost all -regular graphs have
separation dimension at least .Comment: One result proved in this paper is also present in arXiv:1212.675
On combinatorial optimisation in analysis of protein-protein interaction and protein folding networks
Abstract: Protein-protein interaction networks and protein folding networks represent prominent research topics at the intersection of bioinformatics and network science. In this paper, we present a study of these networks from combinatorial optimisation point of view. Using a combination of classical heuristics and stochastic optimisation techniques, we were able to identify several interesting combinatorial properties of biological networks of the COSIN project. We obtained optimal or near-optimal solutions to maximum clique and chromatic number problems for these networks. We also explore patterns of both non-overlapping and overlapping cliques in these networks. Optimal or near-optimal solutions to partitioning of these networks into non-overlapping cliques and to maximum independent set problem were discovered. Maximal cliques are explored by enumerative techniques. Domination in these networks is briefly studied, too. Applications and extensions of our findings are discussed
The Minrank of Random Graphs
The minrank of a graph is the minimum rank of a matrix that can be
obtained from the adjacency matrix of by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
for all regimes of . In particular, for any constant ,
we show that with high probability,
where is chosen from . This bound gives a near quadratic
improvement over the previous best lower bound of (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
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