The sum number of the cocktail party graph

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a finite number of isolated vertices to it will always yield a sum graph, and the sum number oe(G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K oe(G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that oe(H 2;n ) = 4n \Gamma 5, and in the general case we give constructive proofs that oe(H m;n ) 2 \Omega\Gamma mn) and oe(H m;n ) 2 O(mn 2 ). We conjecture that oe(H m;n ) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the first known graph with this property. We also provide for the first time an optimal labelling of the complete bipatite graph Kmn whose smallest label is 1

Laboratoire, le temple et le marché : élargir le débat

Version anglaise disponible dans la Bibliothèque numérique du CRDI : Lab, the temple, and the market : expanding the conversationTitres apparentés: Lab, the temple, and the market : reflections at the intersection of science, religion, and development (116242), et; Labo, le temple et le marché : réflexions sur le rôle de la science et de la religion au regard du développement (114491

Lab, the temple, and the market : expanding the conversation

French version available in IDRC Digital Library: Laboratoire, le temple et le marché : élargir le débatRelated titles: Lab, the temple, and the market : reflections at the intersection of science, religion, and development (116242), and; Lab, the temple and the market : reflections on the role of science and religion in development (114490

The sum number of a disjoint union of graphs

In this paper we consider the disjoint union of graphs as sum graphs. We provide an upper bound on the sum number of a disjoint union of graphs and provide an application for the exclusive sum number of a graph. We conclude with some open problems

Labelling wheels for minimum sum number

A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) +L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number oe(H) of a connected graph H is the least number r of isolated vertices K r such that G = H+K r is a sum graph. It is clear that if H is of size m, then oe(H) m. Recently Hartsfield and Smyth showed that for wheels W n of order n+1 and size m = 2n, oe(W n ) 2 Theta(m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we refine these results to show that for even n 4, oe(W n ) = n=2 + 2, while for odd n 5 we disprove a conjecture of Hartsfield and Smyth by showing that oe(W n ) = n. Labellings are given that achieve these minima

Indeterminate strings, prefix arrays & undirected graphs

An integer array y=y[1..n] is said to be feasible if and only if y[1]=n and, for every i∈2..n, i≤i+y[i]≤n+1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet Σ; otherwise it is said to be regular. A feasible array y is said to be regular if and only if it is the prefix array of some regular string. We show using a graph model that every feasible array of integers is a prefix array of some (indeterminate or regular) string, and for regular strings corresponding to y, we use the model to provide a lower bound on the alphabet size. We show further that there is a 1–1 correspondence between labelled simple graphs and indeterminate strings, and we show how to determine the minimum alphabet size σ of an indeterminate string x based on its associated graph Gx. Thus, in this sense, indeterminate strings are a more natural object of combinatorial interest than the strings on elements of Σ that have traditionally been studied

Verifying a border array in linear time

A border of a string x is a proper (but possibly empty) prefix of x that is also a suffix of x. The border array β = β[1..n] of a string x = x[1..n] is an array of nonnegative integers in which each element β[i], 1 ≤ i ≤ n, is the length of the longest border of x[1..i]. In this paper we first present a simple linear-time algorithm to determine whether or not a given array y = y[1..n] of integers is a border array of some string on an alphabet of unbounded size. We state as an open problem the design of a corresponding and equally efficient algorithm on an alphabet of bounded size α. We then consider the problem of generating all possible distinct border arrays of given length n on a bounded or unbounded alphabet, and doing so in time proportional to the number of arrays generated. A previously published algorithm that claims to solve this problem in constant time per array generated is shown to be incorrect, and new algorithms are proposed. We state as open the design of an equally efficient on-line algorithm for this problem

Has the evolution of complexity in the amphibian papilla influenced anuran speciation rates?

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72375/1/j.1420-9101.2006.01079.x.pd

A critical comparison of integral projection and matrix projection models for demographic analysis

Structured demographic models are among the most common and useful tools in population biology. However, the introduction of integral projection models (IPMs) has caused a profound shift in the way many demographic models are conceptualized. Some researchers have argued that IPMs, by explicitly representing demographic processes as continuous functions of state variables such as size, are more statistically efficient, biologically realistic, and accurate than classic matrix projection models, calling into question the usefulness of the many studies based on matrix models. Here, we evaluate how IPMs and matrix models differ, as well as the extent to which these differences matter for estimation of key model outputs, including population growth rates, sensitivity patterns, and life spans. First, we detail the steps in constructing and using each type of model. Second, we present a review of published demographic models, concentrating on size-based studies, which shows significant overlap in the way IPMs and matrix models are constructed and analyzed. Third, to assess the impact of various modeling decisions on demographic predictions, we ran a series of simulations based on size-based demographic data sets for five biologically diverse species. We found little evidence that discrete vital rate estimation is less accurate than continuous functions across a wide range of sample sizes or size classes (equivalently bin numbers or mesh points). Most model outputs quickly converged with modest class numbers (≥10), regardless of most other modeling decisions. Another surprising result was that the most commonly used method to discretize growth rates for IPM analyses can introduce substantial error into model outputs. Finally, we show that empirical sample sizes generally matter more than modeling approach for the accuracy of demographic outputs. Based on these results, we provide specific recommendations to those constructing and evaluating structured population models. Both our literature review and simulations question the treatment of IPMs as a clearly distinct modeling approach or one that is inherently more accurate than classic matrix models. Importantly, this suggests that matrix models, representing the vast majority of past demographic analyses available for comparative and conservation work, continue to be useful and important sources of demographic information.Support for this work was provided by NSF awards 1146489, 1242558, 1242355, 1353781, 1340024, 1753980, and 1753954, 1144807, 0841423, and 1144083. Support also came from USDA NIFA Postdoctoral Fellowship (award no. 2019-67012-29726/project accession no. 1019364) for R. K. Shriver; the Swiss Polar Institute of Food and Agriculture for N. I. Chardon; the ICREA under the ICREA Academia Programme for C. Linares; and SERDP contract RC-2512 and USDA National Institute of Food and Agriculture, Hatch project 1016746 for A .M. Louthan. This is Contribution no. 21-177-J from the Kansas Agricultural Experiment Station