The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde

With the Coxeter-diagram conventions of Coxeter diagrams: edges, labels, components and finite type (a finite simple graph with edge labels in $\{3,4,5,\dots \}\cup \{\mathrm{\infty }\}$, label $3$ omitted), the standard affine diagrams are the following labelled graphs. Within each clause, distinct vertex symbols denote distinct vertices; pairs not listed as edges are nonedges and have Coxeter label $2$.

(1) The A family. ${\tilde{A}}_1$ is the graph with two vertices joined by one edge labelled $\mathrm{\infty }$. For $n\ge 2$, ${\tilde{A}}_n$ is the cycle $v_0,v_1,\dots ,v_n$ on $n+1$ vertices, with edges $\{v_i,v_{i+1}\}$ for $0\le i<n$ and $\{v_n,v_0\}$, all labelled $3$ (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges). Thus the family "${\tilde{A}}_n$, $n\ge 1$" is ${\tilde{A}}_1$ for $n=1$ and the $(n+1)$-cycle for $n\ge 2$.

(2) The B family. For $n\ge 3$, ${\tilde{B}}_n$ has vertices $v_0,v_1,\dots ,v_n$ and edges $\{v_0,v_2\}$ and $\{v_1,v_2\}$ with label $3$, together with the path edges $\{v_i,v_{i+1}\}$ for $2\le i\le n-1$, where $\{v_{n-1},v_n\}$ carries the label $4$ and every other edge the label $3$. Thus ${\tilde{B}}_n$ is the path $v_1-\cdots -v_n$ with the $B_n$ labels $3,\dots ,3,4$, with one extra vertex $v_0$ attached to $v_2$ by a $3$-edge. (The recipe is stated for $n\ge 3$; for $n=2$ the family is defined by the convention ${\tilde{B}}_2:={\tilde{C}}_2$ of (7), the three-vertex path with both edges labelled $4$.)

(3) The C family. For $n\ge 2$, ${\tilde{C}}_n$ is the path on $n+1$ vertices $v_0-v_1-\cdots -v_n$ with label $4$ on the two end edges $\{v_0,v_1\}$ and $\{v_{n-1},v_n\}$ and label $3$ on all other edges.

(4) The D family. ${\tilde{D}}_4$ is the star with one centre and four leaves (four arms of length $1$), all labels $3$. For $n\ge 5$, ${\tilde{D}}_n$ has two branch vertices $a,b$ joined by a chain $a=w_0,w_1,\dots ,w_{n-4}=b$ with exactly $n-4$ edges, with two leaves attached to $a$ and two leaves attached to $b$; all labels $3$. (The chain has $n-3$ vertices, so the total number of vertices is $(n-3)+4=n+1$.)

(5) The E family. A star with arms $(p,q,r)$ is a tree with one vertex of degree $3$ and three paths (arms) of $p$, $q$, $r$ edges from it (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree, Connected graphs and connected components defined by the existence of vertex paths). Then ${\tilde{E}}_6$ is the star with arms $(2,2,2)$ (seven vertices), ${\tilde{E}}_7$ the star with arms $(1,3,3)$ (eight vertices), and ${\tilde{E}}_8$ the star with arms $(1,2,5)$ (nine vertices); all labels are $3$.

(6) The F and G families. ${\tilde{F}}_4$ is the path on five vertices with labels $(3,3,4,3)$; ${\tilde{G}}_2$ is the path on three vertices with labels $(3,6)$.

(7) Coincidences. The convention ${\tilde{B}}_2:={\tilde{C}}_2$ gives the three-vertex path with both edges labelled $4$. The finite diagrams $B_2$, $C_2$ and $I_2(4)$ are each the two-vertex graph with one edge labelled $4$; their coincidence as finite Coxeter systems is Classification of finite Coxeter systems, including the H and dihedral families (4). With the naming conventions ${\tilde{C}}_1:={\tilde{A}}_1$, $D_3:=A_3$ (the finite coincidence $A_3=D_3$ of (4) of that theorem), $E_4:=A_4$ and $E_5:=D_5$, one has ${\tilde{D}}_3={\tilde{A}}_3$, ${\tilde{E}}_4={\tilde{A}}_4$ and ${\tilde{E}}_5={\tilde{D}}_5$. Apart from these identifications the diagrams (1)-(6) are pairwise non-isomorphic as labelled graphs (Graph isomorphisms, automorphisms and graph complements); the verification is clause (7) of Crystallographic alcove diagrams: the affine list realized by Weyl types A–G ↗. ${\tilde{A}}_1$ is the only diagram of the list with a label $\mathrm{\infty }$, and every other label in the list lies in $\{3,4,6\}$.

Each recipe gives a finite labelled simple graph. The bounds $n\ge 3$ in the ${\tilde{B}}_n$ recipe and $n\ge 2$ in the ${\tilde{C}}_n$ recipe make the terminal $4$-edge distinct from the branch edges and make the two ${\tilde{C}}_n$ end edges distinct, respectively. The low-rank aliases in (7) complete the family naming. Every vertex and edge is explicitly specified, so no choice principle is used.

The list is presented once, here, so that every later item, the classification theorem and the companion examples page use the same names and the same low-rank conventions. The identifications of (7) are conventions about which family member a diagram belongs to; they are not claims that the corresponding Coxeter systems are isomorphic as abstract Coxeter groups, and no such claim is used on this page.