Coordination sequences for root lattices and related graphs. (English) Zbl 0921.05058
Summary: The coordination sequence \(s_\Lambda{}(k)\) of a graph \(\Lambda{}\) counts the number of its vertices which have distance \(k\) from a given vertex, where the distance between two vertices is defined as the minimal number of bonds in any path connecting them. For a large class of graphs, including in particular the classical root lattices, we present the coordination sequences and their generating functions, summarizing and extending recent results of Conway and Sloane.
MSC:
| 05C99 | Graph theory |
Online Encyclopedia of Integer Sequences:
Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.Coordination sequence for D_4 lattice.
Coordination sequence for E_8 lattice.
Coordination sequence for D_5 lattice.
Coordination sequence for D_6 lattice.
Coordination sequence for D_7 lattice.
Coordination sequence for D_8 lattice.
Coordination sequence for A_4 lattice.
Coordination sequence for A_5 lattice.
Coordination sequence for A_6 lattice.
Coordination sequence for A_7 lattice.
Coordination sequence for A_8 lattice.
Coordination sequence for A_10 lattice.
Coordination sequence for E_7 lattice.
Coordination sequence for E_6 lattice.
Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.
Coordination sequence for G_2 lattice.
Coordination sequence for F_4 lattice.
Coordination sequence for C_4 lattice.
Coordination sequence for C_5 lattice.
Coordination sequence for C_6 lattice.
Coordination sequence for C_7 lattice.
Coordination sequence for C_8 lattice.
Coordination sequence for root lattice B_2.
Coordination sequence for root lattice B_3.
Coordination sequence for root lattice B_4.
Coordination sequence for root lattice B_5.
Coordination sequence for root lattice B_6.
Coordination sequence for root lattice B_7.
Coordination sequence for root lattice B_8.
Coordination sequence for root lattice B_9.
Coordination sequence for root lattice B_10.
Coordination sequence for root lattice B_11.
Coordination sequence for root lattice B_12.
Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.
Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.
Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.
Square array T(n,k) read by antidiagonals: coordination sequence for lattice D_n.
