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Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
multipartite graph: A graph is k-partite if its vertex set can be partitioned into k disjoint sets called color classes in such a way that every edgejoins vertices in two different color classes (see also coloring). A two-partite graph is called bipartite.
Graph theory
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Joanna A. Ellis-Monaghan, Iain Moffatt
The path of length m is denoted Pm, the cycle of length m is denoted Cm. A caterpillar is a tree with the property that the tree obtained by deleting all its leaves (vertices of degree one) is a path. A complete graph on n vertices, Kn, is a simple graph on n vertices in which each pair of vertices is adjacent. A complete bipartite graphKp,q is a simple graph whose vertex set can be partitioned into sets X and Y of size p and q, such that for each x∈X and y∈Y, there is an edge xy, and every edge has an end in X and an end in Y. Similarly a complete multipartite graphKp1,…,pr is a graph whose vertex set can be partitioned into sets of size p1,…,pr such that for any two vertices u and v in different blocks of the partition there is an edge uv, and every edge has its ends in different blocks of the partition. The n-spoke wheel Wn is the graph obtained from the cycle Cn (the rim) and another vertex (the hub) by adding an edge from each vertex on the rim to the hub. The edges incident with the hub are the spokes. A theta-graph is the graph on two vertices with three parallel edges between them. A generalized theta-graph or k-theta graph or k-bridge graph is a graph obtained from the graph on two vertices with k parallel edges by subdividing each edge (creating a vertex of degree two in the middle of it) any number of times. A ladder is a graph Pn×K2. A circular ladder Ln is a graph Cn×K2. A Möbius ladder Mn is a graph obtained from a cycle C2n by adding an edge between every pair of vertices at distance n.
A classification of community detection methods in social networks: a survey
Published in International Journal of General Systems, 2021
S. Souravlas, A. Sifaleras, M. Tsintogianni, S. Katsavounis
In Plantié and Crampes (2013), the authors classify the approaches into six categories, based on the input and output data types being used. For community detection, the input data is represented by three different types of structures: (a) unipartite graph: a graph whose vertices are individuals and whose edges are links connecting the individuals (friends, family, students, etc.), (b) bipartite graph: this graph type is generated when individuals share data like web pages or links. The first set contains individuals, and the second is a set of data being exchanged, (c) multipartite graph: a bipartite graph with disjoint sets. Regularly, it is finally reduced to bipartite graph. The output data is a set of node groups representing communities. Three types are mentioned: (a) Graph partitions, where each node is associated with only one group of nodes and no overlaps occur. (b) Hypergraphs with overlapping communities included, and (c) Concept graphs, with nodes sharing a number of common properties.