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Abstract Island archipelagos in the South Pacific have relatively high species endemism within the insect order Odonata, specifically damselflies. Nesobasis Selys, 1891, an endemic damselfly genus from Fiji, includes over 20 species, but a clear understanding of its evolutionary relationship to other damselflies in the region is lacking. Scientists have questioned the monophyly of Nesobasis due to variations within the genus leading to the establishment of three divisions provisionally named as: comosa-, erythrops- and longistyla- groups. However, Nesobasis has shown to be monophyletic in previous phylogenetic analyses. Using additional species in this study, we investigate the phylogenetic relationships between Nesobasis and other damselflies from the region, specifically the endemic Vanuatubasis Ober & Staniczek, 2009 from the neighboring island archipelagos of Vanuatu. The relationship between these taxa has not yet been examined with molecular data. Five genes were used in a maximum likelihood phylogenetic reconstruction and examined morphological data to determine the relationship between these genera. Our results recover three distinct clades overall with Vanuatubasis nested within Nesobasis (i.e., non-monophyletic). Vanuatubasis is sister to the longistyla and erythrops groups. The third group, comosa, was found sister to the clade of Vanuatubasis ( longistyla + erythrops ). As a result of these findings, we propose the new genus, Nikoulabasis gen. nov. 

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fr ́echet mean. In this work, we equip a set of graph with the pseudometric defined by the l2 norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Fr ́echet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric. 

The availability of large datasets composed of graphs creates an unprecedented need to invent novel tools in statistical learning for graph-valued random variables. To characterize the average of a sample of graphs, one can compute the sample Frechet mean and median graphs. In this paper, we address the following foundational question: does a mean or median graph inherit the structural properties of the graphs in the sample? An important graph property is the edge density; we establish that edge density is an hereditary property, which can be transmitted from a graph sample to its sample Frechet mean or median graphs, irrespective of the method used to estimate the mean or the median. Because of the prominence of the Frechet mean in graph-valued machine learning, this novel theoretical result has some significant practical consequences.