PATH POSITIVITY AND INFINITE COXETER GROUPS

A Coxeter graph is a connected graph each of whose edges is labeled wi th an integer greater-than-or-equal-to 3 or with infinity. The adjacen cy matrix of a Coxeter graph G, denoted by A(G) = (a(ij)), is defined to be a square matrix of order \V\, where a(ij) = 2 cos(pi/p) if the e dge (i, j) is labeled with the integer p, and 0 if there is no edge jo ining vertex i with vertex j. For any positive integer k, we denote by P(k) the characteristic polynomial of the adjacency matrix of the pat h on k vertices. A Coxeter graph G is said to be path-positive if for all positive integers k the matrix P(k)(A(G)) is entrywise nonnegative . It is shown that with the exception of a few cases, which are A, B, D, E, F, H, and I, any Coxeter graph is path-positive. The result can be interpreted as a new criterion for the infiniteness of a Coxeter gr oup.