5 research outputs found
On the unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial
of a graph, including unimodality, log-concavity and reality of zeros. We
establish recurrence relations and give factorizations of independence
polynomials for certain classes of graphs. As applications we settle some
unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of