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130 6 ⋅ software around one bit per edge [Boldi et al., 2009]. Our libbvg code (section 6.5) re-implements pieces of the bvgraph framework in portable C code to enable a Matlab wrapper and a shared memory parallel interface. Writing a true Python wrapper for the graphs should also be possible. Many of the contributions of this thesis have already been published. Each publication has an accompanying software package providing all the source code and experiment scripts. These packages are briefly described in sec- tion 6.6. 6.1 adjacency matrices Throughout this chapter, we work with graphs. Each graph G = (V,E) consists of two sets: V = {1, . . . , n} is a set of vertices and E is a set of edges. Eachedge(u,v)= e ∈E isanorderedpairofverticeswithu ∈V,v ∈V. Each vertex is already identified by a numeric index, and we identify graphs with their binary adjacency matrix: ⎧ ⎪1 (i,j)∈E A=[Aij] Aij =⎨ (6.1) ⎪⎩0 otherwise. An undirected graph has both (i, j) and (j, i) in E. Hence, when G is undirected, then A is symmetric. All of the following generalizations of the binary adjacency matrix maintain this property. We handle graphs with weighted edges in two cases. In both cases, we consider the weights as elements from R. Let w(e) be a map from edges e ∈ E to weights in R. When all the weights exclude the value 0, our first case, then the weighted adjacency matrix is ⎧ ⎪w(e) e=(i,j)∈E A=[Aij] Aij =⎨ (6.2) ⎪⎩0 otherwise.

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