1,243 research outputs found
A Spinorial Formulation of the Maximum Clique Problem of a Graph
We present a new formulation of the maximum clique problem of a graph in
complex space. We start observing that the adjacency matrix A of a graph can
always be written in the form A = B B where B is a complex, symmetric matrix
formed by vectors of zero length (null vectors) and the maximum clique problem
can be transformed in a geometrical problem for these vectors. This problem, in
turn, is translated in spinorial language and we show that each graph uniquely
identifies a set of pure spinors, that is vectors of the endomorphism space of
Clifford algebras, and the maximum clique problem is formalized in this setting
so that, this much studied problem, may take advantage from recent progresses
of pure spinor geometry
Cliques in rank-1 random graphs: the role of inhomogeneity
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous
random graphs, where edge probabilities between vertices are roughly
proportional to the product of their vertex weights. We show that the clique
number is concentrated on at most two consecutive integers, for which we
provide an expression. Interestingly, the order of the clique number is
primarily determined by the overall edge density, with the inhomogeneity only
affecting multiplicative constants or adding at most a
multiplicative factor. For sparse enough graphs the clique number is always
bounded and the effect of inhomogeneity completely vanishes.Comment: 29 page
On hypergraph Lagrangians
It is conjectured by Frankl and F\"uredi that the -uniform hypergraph with
edges formed by taking the first sets in the colex ordering of
has the largest Lagrangian of all -uniform hypergraphs
with edges in \cite{FF}. Motzkin and Straus' theorem confirms this
conjecture when . For , it is shown by Talbot in \cite{T} that this
conjecture is true when is in certain ranges. In this paper, we explore the
connection between the clique number and Lagrangians for -uniform
hypergraphs. As an implication of this connection, we prove that the
-uniform hypergraph with edges formed by taking the first sets in
the colex ordering of has the largest Lagrangian of all
-uniform graphs with vertices and edges satisfying for
Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1312.7529, arXiv:1211.7057, arXiv:1211.6508, arXiv:1311.140
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