CHARACTERISTIC POLYNOMIALS WITH INTEGER ROOTS
Gordon Royle
School of Mathematics & Statistics University of Western Australia
Bert’s Matroid Jamboree Maastricht 2012
GORDON ROYLE AUSTRALIA
GORDON ROYLE PERTH
GORDON ROYLE WHEREEVERYPROSPECTPLEASES ...
GORDON ROYLE ... AND ONLY MAN IS VILE
GORDON ROYLE −c If M = M(G) is graphic, then C(M(G), z) = z PG(z) where PG(z) is the well-known chromatic polynomial of G. ∗ If M = M(G) is cographic, then C(M(G), z) = FG(z) where FG(z) is the (slightly less) well-known flow polynomial of G.
CHARACTERISTIC POLYNOMIAL
If M = (E, r) is a matroid with rank function r then the polynomial X C(M, z) = (−1)|X|zr(E)−r(X) X⊆E is called the characteristic polynomial of M.
GORDON ROYLE ∗ If M = M(G) is cographic, then C(M(G), z) = FG(z) where FG(z) is the (slightly less) well-known flow polynomial of G.
CHARACTERISTIC POLYNOMIAL
If M = (E, r) is a matroid with rank function r then the polynomial X C(M, z) = (−1)|X|zr(E)−r(X) X⊆E is called the characteristic polynomial of M.
−c If M = M(G) is graphic, then C(M(G), z) = z PG(z) where PG(z) is the well-known chromatic polynomial of G.
GORDON ROYLE CHARACTERISTIC POLYNOMIAL
If M = (E, r) is a matroid with rank function r then the polynomial X C(M, z) = (−1)|X|zr(E)−r(X) X⊆E is called the characteristic polynomial of M.
−c If M = M(G) is graphic, then C(M(G), z) = z PG(z) where PG(z) is the well-known chromatic polynomial of G. ∗ If M = M(G) is cographic, then C(M(G), z) = FG(z) where FG(z) is the (slightly less) well-known flow polynomial of G.
GORDON ROYLE EXAMPLES
The complete graph Kn has characteristic polynomial
C(Kn, z) = (z − 1)(z − 2) ... (z − n).
7 In the Fano plane F7 the size/rank of the 2 subsets is given by
|X|\r(X) 0 1 2 3 4 5 6 7 0 1 1 7 2 21 7 3 28 35 21 7 1
3 2 C(F7, z) = z + z (−7) + z(21 − 7) + (−28 + 35 − 21 + 7 − 1) = (z − 1)(z − 2)(z − 4)
GORDON ROYLE BASIC PROPERTIES
For a simple matroid M = (E, r), the characteristic polynomial is monic with degree r(E), has alternating coefficients, |E| has leading coefficients 1, −|E|, 2 − γ3, where γ3 is the number of 3-element circuits of M.
GORDON ROYLE 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 PG(x) > 0 Birkhoff-Lewis Conjecture [still unsolved]
If G is planar and x ∈ (4, 5), then PG(x) > 0. This led to the study of the real chromatic roots of graphs, and then to the complex chromatic roots of graphs.
GORDON ROYLE 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.
GORDON ROYLE UPPERBOUNDS
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 Kk minor.
1Proved for graphs by Woodall in 1992, and for general matroids 15 years earlier by Oxley
GORDON ROYLE UPPERROOT-FREE INTERVALS
Can something analogous be said about minor-closed classes of matroids, or even just binary matroids? A “most-wanted” test case2 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 ...
2that is, most-wanted by me
GORDON ROYLE Mostly, the answer is “Nothing much”, but sometimes a little more can be said.
ALLROOTSINTEGRAL
For all kinds of graphical (and other polynomials) a popular question is:
What can be said when the polynomial has all roots integral?
GORDON ROYLE ALLROOTSINTEGRAL
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.
GORDON ROYLE (z − 2)(z − 3)
Chordal graphs have chromatic polynomials with only integer roots.
CHORDALGRAPHS
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)
GORDON ROYLE (z − 3)
Chordal graphs have chromatic polynomials with only integer roots.
CHORDALGRAPHS
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)
GORDON ROYLE Chordal graphs have chromatic polynomials with only integer roots.
CHORDALGRAPHS
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)
GORDON ROYLE CHORDALGRAPHS
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.
GORDON ROYLE BUTSODOMANYOTHERS ...
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.
GORDON ROYLE PLANARGRAPHS
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
GORDON ROYLE FLOWROOTS
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”.
GORDON ROYLE A planar chordal graph has many separating triangles, so a dual planar chordal graph has lots of 3-edge cutsets.
DUALPLANARCHORDALGRAPHS
GORDON ROYLE A planar chordal graph has many separating triangles, so a dual planar chordal graph has lots of 3-edge cutsets.
DUALPLANARCHORDALGRAPHS
GORDON ROYLE DUALPLANARCHORDALGRAPHS
A planar chordal graph has many separating triangles, so a dual planar chordal graph has lots of 3-edge cutsets.
GORDON ROYLE 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”!
GORDON ROYLE STEP 1
If a polynomial
n n−1 n−2 f (z) = z − a1z + a2z − ...
has real roots then the coefficients are maximised when
f (z) = (z − λ)n
where λ = a1/n is the average of the roots.
all at λ
GORDON ROYLE STEP 1
If a polynomial
n n−1 n−2 f (z) = z − a1z + a2z − ...
has integer roots then the coefficients are maximised when
f (z) = (z − bλc)δ(z − dλe)n−δ
where λ = a1/n is the average of the roots.
some at bλc and rest at dλe
GORDON ROYLE STEP 2
As the flow polynomial is
|E| C(M, z) = zr − |E|zr−1 + − γ zr−2 − ... 2 3
an upper bound on
|E| − γ 2 3
gives a lower bound on γ3.
GORDON ROYLE STEP 2
After some slightly fiddly details, and lots of coffee
we conclude that γ3 is strictly larger than the number of vertices of degree 3 in G, and so G has a proper 3-edge cutset.
GORDON ROYLE FINALSTEP
A flow analogue of the clique cutset formula: F (z)F (z) F (z) = H J G (z − 1)(z − 2)
G
H J
GORDON ROYLE FINAL STEP
By induction, both H and J are dual planar chordal graphs, and therefore so is G.
GORDON ROYLE FINALREMARKS
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
GORDON ROYLE FINALQUESTION
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(K5)-minor.
GORDON ROYLE Thanks for listening! Hartelijk dank, Bert!
GORDON ROYLE