The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C 1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the C r (r 1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C 1 but not neatly embedded, the other in which S is not of class C 1 .
In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity n − 2, n − 3, n − 4, n − 5 respectively.
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
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