Connected matroids with the smallest whitney numbers

Abstract

In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r( G) = n, and if X ( G) = Σ μ(0, x) λ r − r( x) = Σ (−1) r − k W k λ k is the characteristic polynomial of G, then w k⩾ r k +n r−1 k Thus γ( G) ⩾ 2 r − 1 ( n+2), where γ( G) = Σ w k . In this paper we sharpen these lower bounds for connected geometries: If G is connected, r( G) ⩾ 3, and n( G) ⩾ 2 (( r, n) ≠ (4,3)), then w i⩾ r i + n r i+1 for i>1; w 1⩾r+n r 2 − 1; | μ| ⩾ ( r− 1) n; and γ ⩾ (2 r − 1 − 1)(2 n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an ( n + 1)point line. If G is any series-parallel network, r(G) = r( G ̄ ) = 4 , and n(G) = n( G ̄ ) = 3 then (w 1(G)) 4 t-G ⩾ (w 1( G ̄ )) = (8, 20, 18, 7, 1) . Further, if β is the Crapo invariant, β(G)= d X(G) dλ (1) , then β( G) ⩾ max(1, n − r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.