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On Zeros of the Characteristic Polynomial of Matroids of Bounded Tree-Width

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On Zeros of the Characteristic Polynomial of Matroids of Bounded Tree-Width On zeros of the characteristic polynomial of matroids of bounded tree-width Carolyn Chun∗3, Rhiannon Hally1, Criel Merinoz2 and Steven Noblex1 1Department of Mathematical Sciences, Brunel University London, Uxbridge, UB8 3PH, United Kingdom 2Instituto de Matem´aticas,Universidad Nacional Aut´onomade M´exico, M´exicoCity, M´exico 3Mathematics Department, United States Naval Academy, Annapolis, MD, United States of America August 28, 2016 Abstract We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k, the characteristic polynomial of any loopless, GF (q)- representable matroid with tree-width k has no real zero greater than qk−1. 1 Introduction For a graph G, the chromatic polynomial χG(λ) is an invariant which counts the number of proper colourings of G when evaluated at a non-negative inte- ger λ. However, the chromatic polynomial has an additional interpretation as the zero-temperature antiferromagnetic Potts model of statistical mechan- ics. This has motivated research into the zeros of the chromatic polynomial ∗[email protected] [email protected] [email protected]. Investigaci´on realizada gracias al Programa UNAM- DGAPA-PAPIIT IN102315. [email protected] 1 by theoretical physicists as well as mathematicians. Traditionally, the focus from a graph theory perspective has been the positive integer roots, which correspond to the graph not being properly colourable with λ colours. A growing body of work has begun to emerge in recent years more concerned with the behaviour of real or complex roots of the chromatic polynomial. Sokal [19] proved that the set of roots of chromatic polynomials is dense in the complex plane. In contrast, many other results show that certain regions are free from zeros. For planar graphs, the Birkhoff–Lewis theorem states that the interval [5; 1) is free from zeros. For more results along these lines, we direct the reader to the work of Borgs [1], Jackson [9], Sokal [18], Thomassen [20] and Woodall [21]. Perhaps one of the most compelling open questions concerning real zeros is to determine tight bounds on the largest real zero of the chromatic polynomial. One such bound is given in [18] and depends on the maximum vertex degree. For recent surveys see [17] and [4]. In matroids, the corresponding invariant is the characteristic polynomial. The characteristic polynomial of a loopless matroid M, with ground set E and rank function r, is defined by X r(E)−r(F ) χM (λ) = µM (;;F )λ ; F 2L where L denotes the lattice of flats of M and µM the M¨obiusfunction of L. When M has a loop, χM (λ) is defined to be zero. Observe that for a loopless matroid M, χM (λ) is monic of degree r(E) and that M and its simplification have the same characteristic polynomial. The projective geometry of rank r over GF (q) is denoted by PG(r−1; q), and Ur;n, where n ≥ r, denotes the uniform matroid with rank r containing n elements. In the uniform matroid, every set of r or fewer elements is independent. The characteristic polynomials of PG(r − 1; q) and Ur;n play important roles in this paper, and these are easily computed. For a prime power q, the projective geometry PG(r − 1; q) has lattice of flats isomorphic to the lattice of subspaces of the r-dimensional vector space over GF (q). Hence it has characteristic polynomial 2 r−1 χPG(r−1;q)(λ) = (λ − 1)(λ − q)(λ − q ) ··· (λ − q ): (1) The largest root of the characteristic polynomial for a projective geometry is r−1 therefore q . The characteristic polynomial of the uniform matroid, Ur;n, is r−1 X k n r−k χU (λ) = (−1) (λ − 1): r;n k k=0 2 For more background on matroid theory, we suggest that the reader consults [15]. For the theory of the M¨obiusfunction and the characteristic polynomial, we recommend [3, 22]. Perhaps the most compelling open question concerning real zeros in this context is deciding whether there is an upper bound for the real roots of the characteristic polynomial of any matroid belonging to a specified minor- closed class. Welsh conjectured that no cographic matroid has a charac- teristic polynomial with a root in (4; 1). This was recently disproved by Haggard et al. in [7], and, in [10], Jacobsen and Salas showed that there are cographic matroids whose characteristic polynomials have roots exceed- ing five. Consequently, determining whether an upper bound exists for the roots of the characteristic polynomials of cographic matroids remains open. In [17], Royle conjectured that for any minor-closed class of GF (q)- representable matroids, not including all graphs, there is a bound on the largest real root of the characteristic polynomial. Given the situation with cographic matroids, this is clearly a difficult conjecture to resolve in the affir- mative. In contrast, the situation with graphic matroids has been resolved. Thomassen [20] noted that by combining a result that he and Woodall [21] had obtained independently with a result of Mader [12], one obtains the following. Theorem 1.1. Let F be a proper minor-closed family of graphs. Then there exists c 2 R such that the chromatic polynomial of any loopless graph G in F has no root larger than c. For certain minor-closed families of graphs, one can find the best possible constant c. One such example is the class of graphs with bounded tree-width, a concept originally introduced by Robertson and Seymour [16]. A tree- decomposition of a graph G comprises a tree T and a collection fXtgt2V (T ) of subsets of V (G) satisfying the following properties. 1. For every edge uv of G, there is a vertex t of T such that fu; vg ⊆ Xt. 2. If p and r are distinct vertices in T , the vertex v is in Xp \ Xr and q lies on the path from p to r in T , then v 2 Xq. The width of a tree-decomposition is maxt2V (T ) jXtj − 1 and the tree-width of a graph is the minimum width of all of its tree-decompositions. As its name suggests, graph tree-width measures how closely a graph resembles a tree. Matroid tree-width, which we will define later, measures how closely a matroid resembles a tree. If a graph can be obtained by gluing small graphs together in a tree-like structure, then it has small tree-width. Likewise, if a 3 matroid can be obtained by gluing small matroids together along a tree-like pattern, then it has small matroid tree-width. Thomassen [20] proved the following. Theorem 1.2. For positive integer k, let G be a graph with tree-width at most k. Then the chromatic polynomial, χG(λ), is identically zero or else χG(λ) > 0 for all λ > k. Thomassen's proof proceeded essentially as follows, using induction on the number of vertices of G. Let G have tree-width k. Take a tree- decomposition of width k, with notation as above. Choose s and t to be neighbouring vertices in T . Then Xs \ Xt is a vertex-cut of G. One may add edges to G with both end-vertices in Xs \ Xt until Xs \ Xt forms a clique without altering the tree-width. Call this new graph G0. The chro- matic polynomial of G may be written in terms of the chromatic polynomial of graphs with fewer vertices than G having tree-width at most k and the chromatic polynomial of G0 in such a way that one may apply induction provided the result can be established for G0. But since G0 has a clique whose vertices comprise a vertex-cut, the chromatic polynomial of G0 may also be expressed in terms of the chromatic polynomials of graphs with fewer vertices and having tree-width at most k. In this paper, we generalize Thomassen's useful technique to matroids. The GF (q)-representable matroid analogue of a clique is a projective geom- etry over GF (q). A given simple graph G sits inside a clique on V (G) in the same way that a simple GF (q)-representable matroid M with rank r sits inside PG(r − 1; q). In the above technique, edges are added to an \area" of G to form a clique restriction, so that the altered graph has a clique vertex-cut. This can be viewed as adding edges from the clique on V (G) to the graph G to obtain a clique, across which our graph may be broken. In this paper, we show how to add elements from PG(r − 1; q) to a certain \area" of M in order to get a GF (q)-representable matroid with a certain projective geometry restriction, across which our matroid may be broken. The map that we use to break apart a matroid is a tree-decomposition, which was established by Hlin˘en´yand Whittle in [8]. They developed a matroid analogue of graph tree-width, which we define formally in Section 3. In order to generalize Thomassen's technique, we first develop some tools for GF (q)-representable matroids of bounded matroid tree-width. We then apply these tools to extend his argument to matroid tree-width, which we shall refer to simply as tree-width when the context is clear. In this way, we demonstrate the utility of these tools and simultaneously make progress towards Royle's conjecture. 4 An alternative way to prove Theorem 1.2 is to combine the observation that every graph with tree-width at most k has a vertex of degree at most k with Lemma 4.2 below, established by Oxley for matroids and rediscovered for the special case of graphs by Thomassen [20] and Woodall [21].
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