Abstract
The most central notion of this thesis is the graph. For reasons that will become clear later, we will not resort to the traditional definition of a graph as a set of vertices and a set of edges. Our definition includes so called multigraphs, graphs where multiple edges are allowed, in a natural way. We will consider sets of half-edges and vertices to be the building blocks of a graph. Based on those sets, a graph consists of a map that associates half-edges with vertices and an involution on the set of half-edges that maps a half-edge to its other half. Naturally, two half-edges make up an edge this way.
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Notes
- 1.
A simple graph is a graph without selfloops or multiple edges between the same pair of vertices.
- 2.
A bipartite graph is a graph, whose vertex set is the union of two disjoints sets of mutually disconnected vertices.
- 3.
These numbers are called the telephone numbers [5, Example II.13].
- 4.
Equivalently, \(\mathfrak {G}_{m,k}\) is the quotient of \(\mathfrak {G}^{\text {lab}}_{m,k}\) under the group action \(*\), \(\mathfrak {G}_{m,k} = \mathfrak {G}^{\text {lab}}_{m,k}/P_{m,k}\).
- 5.
Arguably, it would be clearer to use a map \(\pi \) that maps an arbitrary graph to its unique isomorphic representative in \(\mathfrak {G}\). The product would then read, \(m (\Gamma _1 \otimes \Gamma _2) = \pi ( \Gamma _1 \sqcup \Gamma _2)\). We will omit this map \(\pi \) to agree with the notation commonly used in the literature.
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Borinsky, M. (2018). Graphs. In: Graphs in Perturbation Theory. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-03541-9_2
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DOI: https://doi.org/10.1007/978-3-030-03541-9_2
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