Bapat, R. B. ; Lal, A. K.
(1994)
*Path positivity and infinite Coxeter groups*
Linear Algebra and its Applications, 196
.
pp. 19-35.
ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0024-3795(94)90313-1

## Abstract

A Coxeter graph is a connected graph each of whose edges is labeled with an integer ≥ 3 or with ∞. The adjacency 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(π/p) if the edge (i, j) is labeled with the integer p, and 0 if there is no edge joining vertex i with vertex j. For any positive integer k, we denote by P_{k} the characteristic polynomial of the adjacency matrix of the path 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 group.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

ID Code: | 78319 |

Deposited On: | 19 Jan 2012 06:32 |

Last Modified: | 19 Jan 2012 06:32 |

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