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LiorKogan authored Nov 29, 2024
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@@ -87,7 +87,7 @@ The term _property graph_ refers to both a mathematical structure and a data mod

A [_graph_](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)) is an ordered quintet 𝐺 = (𝑉, 𝐸, 𝐴, _Οˆβ‚‘_, _Οˆβ‚_) consisting of three pairwise disjoint sets and two functions. 𝑉 is a nonempty set whose elements are called [_vertices_](https://en.wikipedia.org/wiki/Vertex_(graph_theory)) (_nodes_, _dots_, _points_), 𝐸 is a set whose elements are called _undirected edges_ (_undirected links_, _undirected lines_), 𝐴 is a set whose elements are called _directed edges_ (_directed links_, _directed lines_, _arcs_, _arrows_), _Οˆβ‚‘: E β†’ { {u,v}: u,v ∈ V }_ is a total function mapping each undirected edge to an unordered pair of vertices, and _Οˆβ‚: A β†’ { (u,v): u,v ∈ V }_ is a total function mapping each directed edge to an ordered pair of vertices. An [_undirected graph_](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Graph) is a graph in which 𝐴 ≔ _βˆ…_. A [_directed graph_](https://en.wikipedia.org/wiki/Directed_graph) (_digraph_, _oriented graph_) is a graph in which 𝐸 ≔ _βˆ…_. A [_mixed graph_](https://en.wikipedia.org/wiki/Mixed_graph) is a graph where both directed and undirected edges may exist.

Given undirected edge 𝑒 such that _Οˆβ‚‘_(𝑒) = {𝑒,𝑣}, we say that 𝑒 is an edge _between_ 𝑒 and 𝑣, 𝑒 _connects_ (_joins_) 𝑒 and 𝑣, and 𝑒 and 𝑣 are _adjacent_. Likewise, given directed edge π‘Ž such that _Οˆβ‚_(π‘Ž) = (𝑒,𝑣), we say that π‘Ž is an edge _from_ 𝑒 _to_ 𝑣, π‘Ž _connects_ (_joins_) 𝑒 to 𝑣, 𝑣 is _adjacent from_ (_out-adjacent to_) 𝑒, 𝑒 is _adjacent to_ (_in-adjacent to_) 𝑣, 𝑒 is the _tail_ (_source vertex_, _initial vertex_) of π‘Ž, and 𝑣 is the _head_ (_target vertex_, _terminal vertex_) of π‘Ž. A [_loop_](https://en.wikipedia.org/wiki/Loop_(graph_theory)) (_self-edge_, _self-loop_, _buckle_) is an undirected or directed edge connecting a vertex with itself - _Οˆβ‚‘_(𝑒) = {𝑒,𝑒} or _Οˆβ‚_(π‘Ž) = (𝑒,𝑒). [_Multiple edges_](https://en.wikipedia.org/wiki/Multiple_edges) (_parallel edges_) are two or more undirected edges connecting the same unordered pair of vertices or directed edges connecting the same ordered pair of vertices. A _simple graph_ is a graph in which loops and multiple edges are not allowed. A [_pseudograph_](http://mathworld.wolfram.com/Pseudograph.html) is a graph in which loops and multiple edges are allowed.
Given undirected edge 𝑒: _Οˆβ‚‘_(𝑒) = {𝑒,𝑣}, we say that 𝑒 is an edge _between_ 𝑒 and 𝑣, 𝑒 _connects_ (_joins_) 𝑒 and 𝑣, and 𝑒 and 𝑣 are _adjacent_. Likewise, given directed edge π‘Ž: _Οˆβ‚_(π‘Ž) = (𝑒,𝑣), we say that π‘Ž is an edge _from_ 𝑒 _to_ 𝑣, π‘Ž _connects_ (_joins_) 𝑒 to 𝑣, 𝑣 is _adjacent from_ (_out-adjacent to_) 𝑒, 𝑒 is _adjacent to_ (_in-adjacent to_) 𝑣, 𝑒 is the _tail_ (_source vertex_, _initial vertex_) of π‘Ž, and 𝑣 is the _head_ (_target vertex_, _terminal vertex_) of π‘Ž. A [_loop_](https://en.wikipedia.org/wiki/Loop_(graph_theory)) (_self-edge_, _self-loop_, _buckle_) is an undirected edge 𝑒: _Οˆβ‚‘_(𝑒) = {𝑒,𝑒} or a directed edge π‘Ž: _Οˆβ‚_(π‘Ž) = (𝑒,𝑒) connecting a vertex _with_ itself. [_Multiple edges_](https://en.wikipedia.org/wiki/Multiple_edges) (_parallel edges_) are two or more undirected edges connecting the same unordered pair of vertices or directed edges connecting the same ordered pair of vertices. A _simple graph_ is a graph in which loops and multiple edges are not allowed. A [_pseudograph_](http://mathworld.wolfram.com/Pseudograph.html) is a graph in which loops and multiple edges are allowed.

An _attributed graph_ is a generic term referring to graphs in which an attribute (_single-attributed graph_) or a set, a bag, or a list of attributes (_multi-attributed graph_) may be associated with each vertex (_vertex-attributed graph_), edge (_edge-attributed graph_), or the graph itself. An _attribute_ may be a nominal value, an ordinal value, a key-value pair, or other annotation. (The term _attributes_ is sometimes used to refer only to key-value pairs, while _labels_ is used to refer to nominal attributes).

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