edge-centric graph

the following definition is an edge-centric definition of graphs, in which edges are primary and vertices are defined in terms of edges.

for any finite set $\text{[math]}$ , a graph on $\text{[math]}$ is a pair $\text{[math]}$ where $\text{[math]}$ is a broken link: blk:def-partition of the set $\text{[math]}$ , called the dart set of $\text{[math]}$ . that is, $\text{[math]}$ is a collection of disjoint, nonempty, mutually exhaustive subsets of $\text{[math]}$ . each subset is a vertex of $\text{[math]}$ . for any $\text{[math]}$ , the darts of $\text{[math]}$ are the pairs $\text{[math]}$ and $\text{[math]}$ , of which the primary dart of $\text{[math]}$ is $\text{[math]}$ . for brevity, we can write $\text{[math]}$ as $\text{[math]}$ and $\text{[math]}$ as $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 2.1 edge-centric definition of graphs]

the head of a dart $\text{[math]}$ is the block $\text{[math]}$ such that $\text{[math]}$ contains $\text{[math]}$ . the tail of $\text{[math]}$ is the head of $\text{[math]}$ . note that this isnt exactly synonymous with edge direction, an edge $\text{[math]}$ can be directed towards $\text{[math]}$ but we can have a dart $\text{[math]}$ whose head is $\text{[math]}$ (when $\text{[math]}$ we have $\text{[math]}$ ).
[cite:;taken from @graph_klein_2024 chapter 2.1 edge-centric definition of graphs]

there is one seeming disadvantage: this definition of graphs does not permit the existence of isolated vertices, vertices with no incident edges. this disadvantage is mitigated by interpreting a subset of edges of a graph as a kind of subgraph.
[cite:;taken from @graph_klein_2024 chapter 2 basic graph definitions]

broken link: xopp-figure:/home/mahmooz/brain/pen/1738710096.9678388.xopp

define the bijection rev on darts by $\text{[math]}$ . for a dart $\text{[math]}$ , $\text{[math]}$ is called the reverse of $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 2 basic graph definitions]

the terminology dart reverse may be deceiving, consider the edges $\text{[math]}$ , their corresponding darts are $\text{[math]}$ , we have that $\text{[math]}$ but $\text{[math]}$ because $\text{[math]}$ .

Let $\text{[math]}$ be a graph. the dart space of $\text{[math]}$ is $\text{[math]}$ , the set of vectors a that assign a real number $\text{[math]}$ to each dart $\text{[math]}$ . for a vector $\text{[math]}$ in dart space and given a set $\text{[math]}$ of darts, $\text{[math]}$ denotes $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.3 vector spaces]

the vertex space of $\text{[math]}$ is $\text{[math]}$ . a vector of vertex space is called a vertex vector.
[cite:;taken from @graph_klein_2024 chapter 3.3 vector spaces]

the arc space of $\text{[math]}$ is a vector subspace of the dart space, namely the set of vectors $\text{[math]}$ in the dart space that satisfy antisymmetry:
$\text{[math]}$ a vector in arc space is called an arc vector.
[cite:;taken from @graph_klein_2024 chapter 3.3 vector spaces]

for a dart $\text{[math]}$ , define $\text{[math]}$ to be the arc vector such that $\text{[math]}$ and $\text{[math]}$ for all darts $\text{[math]}$ such that $\text{[math]}$ and $\text{[math]}$ . we extend this notation to sets of darts: $\text{[math]}$ .
formally, a vertex $\text{[math]}$ is the set of darts having $\text{[math]}$ as head, so $\text{[math]}$ where the sum is over those darts whose heads are $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.3 vector spaces]

for a graph $\text{[math]}$ , we denote by $\text{[math]}$ the dart-vertex incidence matrix, the matrix whose columns are the vectors $\text{[math]}$ . that is, $\text{[math]}$ has a row for each dart $\text{[math]}$ and a column for each vertex $\text{[math]}$ , and the $\text{[math]}$ entry is 1 if $\text{[math]}$ is the head of $\text{[math]}$ if $\text{[math]}$ is the tail of $\text{[math]}$ , and zero otherwise.
[cite:;taken from @graph_klein_2024 chapter 3.3 vector spaces]

[cite:;found in @graph_klein_2024 chapter 3.3 vector spaces]