Structural properties of the minimum cut of partially-supplied graphs.

It is well known that information about the structure of a graph is contained within its minimum cut. Here we investigate how the minimum cut of one graph informs the structure of a second, related graph. We consider pairs of graphs G and H, with respective Laplacian matrices L and M, and call H partially supplied provided M is a Schur complement of L. Our results show how the minimum cut of H relates to the structure of the larger graph G.

An eigenvector interlacing property of graphs that arise from trees by Schur complementation of the Laplacian.

The literature is replete with rich connections between the structure of a graph G = (V, E) and the spectral properties of its Laplacian matrix L. This paper establishes similar connections between the structure of G and the Laplacian L* of a second graph G*. Our interest lies in L* that can be obtained from L by Schur complementation, in which case we say that G* is partially-supplied with respect to G. In particular, we specialize to where G is a tree with points of articulation r ∈ R and consider the partially-supplied graph G* derived from G by taking the Schur complement with respect to R in L. Our results characterize how the eigenvectors of the Laplacian of G* relate to each other and to the structure of the tree.

Experimental evaluation of evolution and coevolution as agents of ecosystem change in Trinidadian streams.

Evolution has been shown to be a critical determinant of ecological processes in some systems, but its importance relative to traditional ecological effects is not well known. In addition, almost nothing is known about the role of coevolution in shaping ecosystem function. Here, we experimentally evaluated the relative effects of species invasion (a traditional ecological effect), evolution and coevolution on ecosystem processes in Trinidadian streams. We manipulated the presence and population-of-origin of two common fish species, the guppy (Poecilia reticulata) and the killifish (Rivulus hartii). We measured epilithic algal biomass and accrual, aquatic invertebrate biomass, and detrital decomposition. Our results show that, for some ecosystem responses, the effects of evolution and coevolution were larger than the effects of species invasion. Guppy evolution in response to alternative predation regimes significantly influenced algal biomass and accrual rates. Guppies from a high-predation site caused an increase in algae relative to guppies from a low-predation site; algae effects were probably shaped by observed divergence in rates of nutrient excretion and algae consumption. Rivulus-guppy coevolution significantly influenced the biomass of aquatic invertebrates. Locally coevolved populations reduced invertebrate biomass relative to non-coevolved populations. These results challenge the general assumption that intraspecific diversity is a less critical determinant of ecosystem function than is interspecific diversity. Given existing evidence for contemporary evolution in these fish species, our findings suggest considerable potential for eco-evolutionary feedbacks to operate as populations adapt to natural or anthropogenic perturbations.