Figure 1

Some graphs used in the discussion of global and local spectral bipartivity measures. (a) Graphs obtained by the addition of one frustrated edge (dotted line) to stars $K_{1,N2}$. Only one triangle is introduced to each graph after the addition of the new links. Thus, the proportion of even to total CWs $(\text{even}+\text{odd})$, $\beta \left(G\right)$, increases as $N_2$ increases and the networks with larger $N_2$ are more bipartite. The proportion of frustrated to total number of edges decreases as $N_2$ increases and $b_1$ follows the same trend that $\beta \left(G\right)$. (b) Values of network bipartivity, $b_1$ and $\beta \left(G\right)$, for the graphs produced by successive addition of frustrated edges to $K_{2,3}$. It is observed that $\beta \left(G\right)$ shows a monotonically decreasing trend from the bipartite graph to the complete graph. $b_1$ does not differentiate between pairs of graphs having the same number of frustrated edges despite the fact that they show different proportions of even to total number of CWs. (c) Values of $\beta \left(i\right)$ and $\beta \left(e\right)$ for the first two graphs in (b). The lowest values of these measures indicate those nodes and/or links which contribute more to the nonbipartivity of the graph. Despite the fact that some links with values different from one are not frustrated edges, their being removed reduces the number of odd CWs, e.g., triangles, increasing the bipartivity of the network.Reuse & Permissions

Figure 2

(Color online). Grassland food web represented as an almost bipartite network. Up-triangles (green) represent grass, (blue) squares represent herbivores, (yellow) circles represent parasitoids, (red) down triangles represent hyperparasitoids, and (pink) diamonds represent hyper-hyperparasitoids. The network bipartivity is illustrated as two sets of nodes situated at the top and bottom of the graphic. Links between nodes in disjointed sets are the vertical ones (black) and those between nodes in the same set are the horizontal ones (red).Reuse & Permissions