N ov 2 00 5 Counting Close Vertices in the Affine Buildings of SL n ( F ) and Sp n ( F )

2 00 5 Counting Close Vertices in the Affine Buildings of SL n ( F ) and Sp n ( F )

For a local field F with finite residue field, we count close vertices in the affine buildings of SLn(F ) and Spn(F ), addressing a question of [4]. In the case of SLn(F ), we give an explicit formula for the number ωn of vertices close to a given vertex. We establish the conjecture following Proposition 3.4 of [4] relating ωn and the number of chambers containing a given vertex. We give analog...

J an 2 00 6 Counting Close Vertices in the Affine Buildings of SL n ( F ) and Sp n ( F )

For a local field F with finite residue field, we count close vertices in the affine buildings of SLn(F ) and Spn(F ), addressing a question of [6]. In the case of SLn(F ), we give an explicit formula for the number ωn of vertices close to a given vertex. We establish the conjecture following Proposition 3.4 of [6] relating ωn and the number of chambers containing a given vertex. We give analog...

1 5 N ov 2 00 2 Representations of SL 2 and the distribution of points in

Distance in the Affine Buildings of SLn and Spn

For a local field K and n ≥ 2, let Ξn and ∆n denote the affine buildings naturally associated to the special linear and symplectic groups SLn(K) and Spn(K), respectively. We relate the number of vertices in Ξn (n ≥ 3) close (i.e., gallery distance 1) to a given vertex in Ξn to the number of chambers in Ξn containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consi...

Further results on odd mean labeling of some subdivision graphs

Let G(V,E) be a graph with p vertices and q edges. A graph G is said to have an odd mean labeling if there exists a function f : V (G) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : E(G) → {1, 3, 5,...,2q - 1} defined by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. A graph that admits an odd mean labelin...