A graph <i>G</i>=(<i>V,E</i>) is representable if there exists a word <i>W</i> over the alphabet <i>V</i> such that letters <i>x</i> and <i>y</i> alternate in <i>W</i> if and only if (<i>x</i> ,<i>y</i>) is in <i>E</i> for each <i>x</i> not equal to <i>y</i> . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for <i>n</i> greater than 3, the line graph of an <i>n</i>-wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for <i>n</i> greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph <i>G</i> which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of <i>G k</i>-times is guaranteed to be non-representable for <i>k</i> greater than 3

In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k = 2, these arrangements correspond to the well-studied Coxeter arrangements. We construct a cell complex P erm k (W ) that is homotopy equivalent to the complement. We then apply discrete Morse theory to obtain a minimal cell complex for the complement. As a result, we give combinatorial interpretations for the Betti numbers, and show that the homology groups are torsion free. We also study a generalization of the Independence Complex of a graph, and show that this generalization is shellable when the graph is a forest. This result is used in studying P erm k (W ) using discrete Morse theory.2000 Mathematics Subject Classification. Primary 05E45.

Abstract. In this paper, we study k-parabolic arrangements, a generalization of k-equal arrangements for finite real reflection groups. When k = 2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement, over C, of the type W Coxeter arrangement is isomorphic to the pure Artin group of type W . Khovanov (1996) gave an algebraic description for the fundamental group of the complement, over R, of the 3-equal arrangement. We generalize Khovanov's result to obtain an algebraic description of the fundamental groups of the complements of 3-parabolic arrangements for arbitrary finite reflection groups. Our description is a real analogue to Brieskorn's description.

A graph G = (V, E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x, y) is in E for each x not equal to y. The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view.In this paper, we prove that for n greater than 3, the line graph of an n-wheel is non-representable. This not only provides a new construction of non-representable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for n greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3.