By Nakanishi N.

ISBN-10: 0677029500

ISBN-13: 9780677029504

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**Extra resources for Graph theory and Feynman integrals**

**Example text**

Let G = (V, E) be a weighted/unweighted digraph; we compute edge connectivity λ(G). We choose an arbitrary vertex s ∈ V as a designated vertex, and we define λ+ s (G) = min{λ(s, v; G) | v ∈ V − s}, λ− s (G) = min{λ(v, s; G) | v ∈ V − s}. 15) Given a minimum cut X of G, λ(G) = min{λ(u, v) | u, v ∈ V (G)} is equal to the λ(u, v) of any u ∈ X and v ∈ V − X . Considering two possible cases s ∈ X and s ∈ V − X , it is immediately shown that it holds: − λ(G) = min{λ+ s (G), λs (G)}. This method therefore computes maximum (s, v)-flows for all v ∈ V − s and maximum (v, s)-flows for all v ∈ V − s, thus running a maximum flow algorithm 2(n − 1) times.

For an undirected (resp. directed) graph G, we denote by E the set of unordered pairs {u, v} such that u, v ∈ V , u = v, and {u, v} ∈ E (resp. ordered pairs (u, v) such that u, v ∈ V , u = v and (u, v) ∈ E) and, for two subsets X, Y ⊆ V (not necessarily disjoint), we define E(X, Y ; G) = {{u, v} ∈ E | u ∈ X, v ∈ Y } (resp. E(X, Y ; G) = {(u, v) ∈ E | u ∈ X, v ∈ Y }), and κ X,Y (G) = min{κ(u, v; G) | {u, v} ∈ E(X, Y ; G)} (resp. κ X,Y (G) = min{κ(u, v; G) | (u, v) ∈ E(X, Y ; G)}), where κ X,Y (G) = +∞ if E(X, Y ; G) = ∅, and we may write E(X, Y ; G) and κ X,Y (G) as E(X, Y ) and κ X,Y , respectively, when G is clear from the context.

Pointer to the next cell. There is also a one-dimensional array that stores the vertex set V (G), where each vertex v in the array is linked to the first cell of Ad j(v) by a pointer. The space for adjacency lists is O( v∈V (G) (1 + d(v; G))) = O(n + m). With adjacency lists, we can find all edges incident with a given vertex v in O(d(v; G)) time by traversing all cells in Ad j(v). To represent a multigraph G, we store an edge set E(v; G) in a linked list Ad j(v), in which vertex u appears |E(v, u; G)| times.

### Graph theory and Feynman integrals by Nakanishi N.

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