CHARACTERISTIC POLYNOMIALS WITH INTEGER ROOTS

Transcription

1 CHARACTERISTIC POLYNOMIALS WITH INTEGER ROOTS Gordon Royle School of Mathematics & Statistics University of Western Australia Bert s Matroid Jamboree Maastricht 2012

2 AUSTRALIA

3 PERTH

4 WHERE EVERY PROSPECT PLEASES...

5 ... AND ONLY MAN IS VILE

6 CHARACTERISTIC POLYNOMIAL If M = (E, r) is a matroid with rank function r then the polynomial C(M, z) = ( 1) X z r(e) r(x) X E is called the characteristic polynomial of M.

7 CHARACTERISTIC POLYNOMIAL If M = (E, r) is a matroid with rank function r then the polynomial C(M, z) = ( 1) X z r(e) r(x) X E is called the characteristic polynomial of M. If M = M(G) is graphic, then C(M(G), z) = z c P G (z) where P G (z) is the well-known chromatic polynomial of G.

8 CHARACTERISTIC POLYNOMIAL If M = (E, r) is a matroid with rank function r then the polynomial C(M, z) = ( 1) X z r(e) r(x) X E is called the characteristic polynomial of M. If M = M(G) is graphic, then C(M(G), z) = z c P G (z) where P G (z) is the well-known chromatic polynomial of G. If M = M(G) is cographic, then C(M(G), z) = F G (z) where F G (z) is the (slightly less) well-known flow polynomial of G.

9 EXAMPLES The complete graph K n has characteristic polynomial C(K n, z) = (z 1)(z 2)... (z n). In the Fano plane F 7 the size/rank of the 2 7 subsets is given by X \r(x) C(F 7, z) = z 3 + z 2 ( 7) + z(21 7) + ( ) = (z 1)(z 2)(z 4)

10 BASIC PROPERTIES For a simple matroid M = (E, r), the characteristic polynomial is monic with degree r(e), has alternating coefficients, has leading coefficients 1, E, ( E ) 2 γ3, where γ 3 is the number of 3-element circuits of M.

11 CHROMATIC ROOTS As C(M, z) is a polynomial, it can be evaluated at any integer, real or complex number, regardless of whether such an evaluation has any combinatorial interpretation. The earliest such result was in the context of chromatic polynomials: Birkhoff-Lewis Theorem (1946) For planar graphs G and real x 5, we have P G (x) > 0 Birkhoff-Lewis Conjecture [still unsolved] If G is planar and x (4, 5), then P G (x) > 0. This led to the study of the real chromatic roots of graphs, and then to the complex chromatic roots of graphs.

12 RESULTS AND CONJECTURES There is a substantial literature on chromatic roots, both real and complex, much of it due to the intimate connection between the chromatic polynomial and the q-state Potts model. In general, we try to answer questions of the form: Are the chromatic roots of a class of graphs absolutely bounded? Are there parameterized bounds in terms of graph parameters? Many fundamental questions remain for chromatic roots, even less is known on flow roots, and almost nothing about characteristic roots of non-graphic, non-cographic matroids.

13 UPPER BOUNDS An upper root-free interval for a family M of matroids is an interval (ρ, ) such that C(M, x) > 0 for all M M, x (ρ, ). Any proper minor-closed class of graphs has an upper root-free interval this follows from two facts: If every simple minor of a matroid has a cocircuit of size at most d then C(M, x) > 0 for all x (d, ), 1 (Mader) There is a function f (k) such that every graph with minimum degree at least f (k) has a K k minor. 1 Proved for graphs by Woodall in 1992, and for general matroids 15 years earlier by Oxley

14 UPPER ROOT-FREE INTERVALS Can something analogous be said about minor-closed classes of matroids, or even just binary matroids? A most-wanted test case 2 is the class of cographic matroids; in other words, bounding the flow roots of graphs. Dominic suggested that perhaps (4, ) is an upper flow-root-free interval I disproved this with graphs with flow roots greater than 4, and suggested that (5, ) is the correct upper flow-root-free interval Statistical physicists Jésus Salas and Jesper Jacobsen disproved this with graphs with flow roots greater than 5, and gave up suggesting anything... 2 that is, most-wanted by me

15 ALL ROOTS INTEGRAL For all kinds of graphical (and other polynomials) a popular question is: What can be said when the polynomial has all roots integral?

16 ALL ROOTS INTEGRAL For all kinds of graphical (and other polynomials) a popular question is: What can be said when the polynomial has all roots integral? Mostly, the answer is Nothing much, but sometimes a little more can be said.

17 CHORDAL GRAPHS A graph is chordal if it can be constructed from a complete graph by repeatedly adding a new vertex adjacent to a clique: (z 1)(z 2)(z 3)

18 CHORDAL GRAPHS A graph is chordal if it can be constructed from a complete graph by repeatedly adding a new vertex adjacent to a clique: (z 1)(z 2)(z 3)(z 2)

19 CHORDAL GRAPHS A graph is chordal if it can be constructed from a complete graph by repeatedly adding a new vertex adjacent to a clique: (z 1)(z 2)(z 3)(z 2)(z 3)

20 CHORDAL GRAPHS A graph is chordal if it can be constructed from a complete graph by repeatedly adding a new vertex adjacent to a clique: (z 1)(z 2)(z 3)(z 2)(z 3) Chordal graphs have chromatic polynomials with only integer roots.

21 BUT SO DO MANY OTHERS... Many non-chordal graphs have integer chromatic roots. Hernández and Luca show that finding similarly structured graphs with integral chromatic roots is equivalent to finding solutions to the Prouhet-Tarry-Escott problem.

22 PLANAR GRAPHS However, if we restrict to planar graphs then all is well: THEOREM (DONG & KOH 1998) A planar graph whose chromatic polynomial has only integer roots is chordal. The proof uses the following ideas: The chromatic polynomial is z(z 1)(z 2) a (z 3) b Counting vertices, edges, faces and triangles shows that either b = 0 or a = 1, b = 1 A result of Whitehead saying that a graph co-chromatic with a 2-tree is a 2-tree

23 FLOW ROOTS Joe Kung and I investigated graphs with integral flow roots. THEOREM (KUNG & ROYLE) A graph with integral flow roots is the planar dual of a planar chordal graph. In other words, the obvious examples are the only examples.

24 DUAL PLANAR CHORDAL GRAPHS

25 DUAL PLANAR CHORDAL GRAPHS

26 DUAL PLANAR CHORDAL GRAPHS A planar chordal graph has many separating triangles, so a dual planar chordal graph has lots of 3-edge cutsets.

27 PROOF IDEAS Suppose M = M(G) is a cographic matroid with integral characteristic roots. Then Use integrality of roots to show that M has lots of 3-circuits, Count things to show that at least one of the 3-circuits is a 3-edge cutset in G, Note that flow polynomials factorize over 3-edge cutsets. Apply induction and, as the old Dutch expression goes, Bert is je oom!

28 STEP 1 If a polynomial f (z) = z n a 1 z n 1 + a 2 z n 2... has real roots then the coefficients are maximised when f (z) = (z λ) n where λ = a 1 /n is the average of the roots. all at λ

29 STEP 1 If a polynomial f (z) = z n a 1 z n 1 + a 2 z n 2... has integer roots then the coefficients are maximised when f (z) = (z λ ) δ (z λ ) n δ where λ = a 1 /n is the average of the roots. some at λ and rest at λ

30 STEP 2 As the flow polynomial is C(M, z) = z r E z r 1 + (( ) ) E γ 3 z r an upper bound on (( ) ) E γ 3 2 gives a lower bound on γ 3.

31 STEP 2 After some slightly fiddly details, and lots of coffee we conclude that γ 3 is strictly larger than the number of vertices of degree 3 in G, and so G has a proper 3-edge cutset.

32 FINAL STEP A flow analogue of the clique cutset formula: F G (z) = F H(z)F J (z) (z 1)(z 2) G H J

33 FINAL STEP By induction, both H and J are dual planar chordal graphs, and therefore so is G.

34 FINAL REMARKS A supersolvable matroid is the matroidal analogue of a chordal graph, and it has integral characteristic roots. For flow roots, what we really showed was two separate things: A cographic matroid with integral characteristic roots is supersolvable A supersolvable cographic matroid is the dual of a planar graph

35 FINAL QUESTION QUESTION Are there other natural classes of (binary) matroids where integral characteristic roots implies supersolvability? Two promising classes to consider: 4-colourable graphs (Dong), and Binary matroids with no M(K 5 )-minor.

36 Thanks for listening! Hartelijk dank, Bert!