## Introduction

Connectivity is a fundamental variable in spatial ecology and biodiversity conservation (e.g. Hanski 1998; Calabrese & Fagan 2004; Crooks & Sanjayan 2006; Kindlmann & Burel 2008), and improving connectivity has been the most prevalent proposed solution for conservation under climate change (Heller & Zavaleta 2009). Therefore, it is no surprise that every year, thousands of publications develop new connectivity-based analyses or apply connectivity as part of an ecologically based analysis. Here, I comment on a branch of connectivity research that has been gaining popularity at an exponential rate during the past decade, that is, graph-theoretic connectivity (Urban & Keitt 2001; Pascual-Hortal & Saura 2006; Minor & Urban 2008; Urban *et al.* 2009). While in year 2000, there were just two citations to graph-theoretic connectivity in ecology and conservation, in 2010, there were >450, with the citation rate doubling approximately every 1.5 years (Fig. 1). With enthusiasm on the rise, it is hard to locate a unified critical perspective about the applicability of graph-theoretic connectivity. Such will be provided here. But, first, what are graphs, really?

Graphs originate from mathematics and computer science. A graph defines relationships between entities, which are frequently called edges and nodes. Graphs are used most naturally when there are relatively few edges between nodes, in which case the graph is easy to visualize and computationally efficient algorithms can be applied to it (Urban *et al.* 2009; Rayfield, Fortin & Fall 2010). Clear visualization, compact standard representation, efficient computation, and centuries-long mathematical background lead to graphs being appealing structures for applied sciences, including ecology. One example of profitable import of graph-theoretic methodology is connectivity motivated by electric circuit theory, where the ecologically relevant feature is that alternative connectivity pathways are accounted for (McRae *et al.* 2008), improving from analysis based on least cost path distances. For more background about graphs, see the comprehensive and well-illustrated review of Newman (2003). An excellent summary from the fields of ecology and conservation is given by Urban *et al.* (2009) and reviews from other biological disciplines by May (2005) and Proulx, Promislow & Phillips (2005).

Integral to the concept of a graph is that it must be possible to define its nodes and edges – something highly relevant below. This is easily carried out in engineering systems, such as energy supply, transportation, or communications networks, where nodes represent supply and demand points and edges represent actual physical connections between nodes. In spatial ecology, the nodes are habitat patches and edges are interpreted as connectivity between patches (Urban & Keitt 2001).