Lattice graph

In <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, a lattice graph, mesh graph, or grid graph is a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> whose <a href="/facts/Graph_drawing/uhh73BVo">drawing</a>, <a href="/facts/Embedding/HKeVKc42">embedded</a> in some <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> ⁠
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
⁠, forms a <a href="/facts/Regular_tiling/tR5CQACy">regular tiling</a>. This implies that the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of <a href="/facts/Bijection/ayC7YE3z">bijective transformations</a> that send the graph to itself is a <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice in the group-theoretical sense</a>.
Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid".
The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the <a href="/facts/Cartesian_product_of_graphs/wC4JJdyU">Cartesian product</a> of a number of <a href="/facts/Complete_graph/AYD6hUqI">complete graphs</a>.

Lattice graph open-in-new

Lattice graph