(Will be extended from time to time.)

**arc**

an edge with a direction

**complete graph**

a complete graph is a graph, in which every vertex is connected to every other vertex

the symbol that is used for a complete graph is *K*_{n}, where n is the number of vertices

**connected**

a net is connected, if there is a continuous path from every vertex to every other vertex

**cycle**

= circuit = closed path;

a cycle is a continuous path (along edges) in which the start and end vertex are the same

**cycles sum**

the sum of two cycles is the set of edges that occur exactly *once* (meaning that the edges that occur twice will be deleted);

example:

**cyclomatic number**

the cyclomatic number *g* is the number of edges that are necessary to complete a spanning tree to make this subgraph a complete connected graph

If there are *v* vertices and *e* edges in the complete connected graph, the cyclomatic number is:

*g* = *e* – *v* +1

**embedding**

a graph/net, in which the vertices have been assigned (Euclidean) coordinates; this implies that now the edges have a certain length

**girth**

the length of the shortest cycle of a graph

if the graph is a tree, the girth is said to be infinite

**graph**

an abstract mathematical object in which *vertices* are connected by *edges*

*k*-connected

an *k*-connected graph is a graph in which *k* numbers of vertices (and their edges) have to be deleted to separate the graph into two parts

graphs, which consists of vertices which are all 3-coordinated can be indeed only 2-connected:

if the two orange vertices were deleted, the graph is *disjoint*

**minimal net**

a net with genus 3

the genus of a surface is – simply speaking – equivalent to the number of its “holes” or “handles”

the primitive unit cell of a 3D-periodic net has at least 3 such holes, for most of the nets the genus is much higher

there are only 8 nets that have these minimal number of holes:

**cds**, **dia, hms, pcu, srs, tfa, tfc,** and **ths**

*n*-coordinated

an *n*-coordinated vertex has *n* connections to other vertices, *n* is the coordination number

shortform: n-c, for instance **nbo** is a 4-c net

if two or more vertices with different coordination numbers are present, the different *coordination numbers* of the vertices are separated by commas and are enclosed in brackets, for instance the net **tbo** is a (3,4)-c net

sometimes, a multinodal net is specified only in its short or compact form, i.e. only the different coordination numbers are given, in other cases every non-symmetry-related vertex is explicitely specified with its coordination number, for instance **zhc** is a octanodal (3,3,3,3,4,4,4,4)-c net; its short notation is again only (3,4)-c;

**net**

a finite or periodic, connected, and simple *graph*

simple means that the edges have *no directions* and that it contains no *loops (*edges linking a vertex to itself), and that there are *no multiple edges* between any vertices

**quasiregular net**

a quasiregular net is characterized by the transitivity *pqrs* = 1112

there is only one quasiregular net:

**fcu**

**regular net**

a regular net is characterized by the transitivity *pqrs* = 1111

there are only 5 regular nets:

**bcu**, **dia**, **nbo**, **pcu**, and **srs**

a regular net is *vertex*– as well as *edge-transitive*

**semiregular net**

a semiregular net is characterized by the transitivity *pqrs* = 11*rs* (with *r* > 1)

there are 16 semiregular nets:

**lvt**, **sod**, **lcs**, **lcv**, **qtz**, **hxg**, **lcy**, **crs**, **bcs**, **acs**, **reo**, **thp**, **rhr**, **ana, ibb, and icc**

a semiregular net is *vertex*– as well as *edge-transitive*

**spanning tree**

a subgraph of a graph, which is a tree and which consists of all vertices of the graph

usually a graph may have several different spanning trees

**tree**

a graph without any closed paths

**uniform net**

in a uniform net at *all* angles the *shortest* rings are all of the same size

Stephen hydeCan you include a definition of a “net”? I recall some discussion about this.

doktorholzPost authoryes, will expand the glossary right after my holiday!

Stephen hydeI was curious to remind myself of the definition… Happy holiday.

Thalles DiogenesHello,

It is a very interesting description. It could be nice to know more about MOF topology. Do you have any suggestion for literature?

Regards,

T.S. Diógenes

doktorholzPost authorHi!

Thanks for your comment!

A general overview about topology of crystalline nets can be found in this chapter and references cited therein:

F. Hoffmann, M. Fröba, in: The Chemistry of Metal-Organic Frameworks: Synthesis, Characterization, and Applications (Ed.: S. Kaskel), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2016, pp. 5–40.

https://onlinelibrary.wiley.com/doi/10.1002/9783527693078.ch2

best wishes!

Frank