CHARACTERISTIC POLYNOMIALS WITH

INTEGER ROOTS

Gordon Royle

School of Mathematics & StatisticsUniversity of Western Australia

Bert’s Matroid JamboreeMaastricht 2012

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AUSTRALIA

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PERTH

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WHERE EVERY PROSPECT PLEASES . . .

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. . . AND ONLY MAN IS VILE

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CHARACTERISTIC POLYNOMIAL

If M = (E, r) is a matroid with rank function r then thepolynomial

C(M, z) =∑X⊆E

(−1)|X|zr(E)−r(X)

is called the characteristic polynomial of M.

If M = M(G) is graphic, then C(M(G), z) = z−cPG(z) wherePG(z) is the well-known chromatic polynomial of G.If M = M(G)∗ is cographic, then C(M(G), z) = FG(z) whereFG(z) is the (slightly less) well-known flow polynomial of G.

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CHARACTERISTIC POLYNOMIAL

If M = (E, r) is a matroid with rank function r then thepolynomial

C(M, z) =∑X⊆E

(−1)|X|zr(E)−r(X)

is called the characteristic polynomial of M.

If M = M(G) is graphic, then C(M(G), z) = z−cPG(z) wherePG(z) is the well-known chromatic polynomial of G.

If M = M(G)∗ is cographic, then C(M(G), z) = FG(z) whereFG(z) is the (slightly less) well-known flow polynomial of G.

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CHARACTERISTIC POLYNOMIAL

If M = (E, r) is a matroid with rank function r then thepolynomial

C(M, z) =∑X⊆E

(−1)|X|zr(E)−r(X)

is called the characteristic polynomial of M.

If M = M(G) is graphic, then C(M(G), z) = z−cPG(z) wherePG(z) is the well-known chromatic polynomial of G.If M = M(G)∗ is cographic, then C(M(G), z) = FG(z) whereFG(z) is the (slightly less) well-known flow polynomial of G.

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EXAMPLES

The complete graph Kn has characteristic polynomial

C(Kn, z) = (z− 1)(z− 2) . . . (z− n).

In the Fano plane F7 the size/rank of the 27 subsets is given by

|X|\r(X) 0 1 2 3 4 5 6 70 11 72 21 73 28 35 21 7 1

C(F7, z) = z3 + z2(−7) + z(21− 7) + (−28 + 35− 21 + 7− 1)

= (z− 1)(z− 2)(z− 4)

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BASIC PROPERTIES

For a simple matroid M = (E, r), the characteristic polynomialis monic with degree r(E),has alternating coefficients,has leading coefficients 1, −|E|,

(|E|2

)− γ3, where γ3 is the

number of 3-element circuits of M.

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CHROMATIC ROOTS

As C(M, z) is a polynomial, it can be evaluated at any integer,real or complex number, regardless of whether such anevaluation has any combinatorial interpretation.

The earliest such result was in the context of chromaticpolynomials:

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, andthen to the complex chromatic roots of graphs.

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RESULTS AND CONJECTURES

There is a substantial literature on chromatic roots, both realand complex, much of it due to the intimate connectionbetween 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 absolutelybounded?Are there parameterized bounds in terms of graphparameters?

Many fundamental questions remain for chromatic roots, evenless is known on flow roots, and almost nothing aboutcharacteristic roots of non-graphic, non-cographic matroids.

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UPPER BOUNDS

An upper root-free interval for a familyM of matroids is aninterval (ρ,∞) such that

C(M, x) > 0 for all M ∈M, x ∈ (ρ,∞).

Any proper minor-closed class of graphs has an upperroot-free interval — this follows from two facts:

If every simple minor of a matroid has a cocircuit of size atmost d then C(M, x) > 0 for all x ∈ (d,∞), 1

(Mader) There is a function f (k) such that every graph withminimum degree at least f (k) has a Kk minor.

1Proved for graphs by Woodall in 1992, and for general matroids 15 yearsearlier by Oxley

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UPPER ROOT-FREE INTERVALS

Can something analogous be said about minor-closed classesof 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 upperflow-root-free intervalI disproved this with graphs with flow roots greater than 4,and suggested that (5,∞) is the correct upperflow-root-free intervalStatistical physicists Jésus Salas and Jesper Jacobsendisproved this with graphs with flow roots greater than 5,and gave up suggesting anything . . .

2that is, most-wanted by meGORDON ROYLE

ALL ROOTS INTEGRAL

For all kinds of graphical (and other polynomials) a popularquestion is:

What can be said when the polynomial has all roots integral?

Mostly, the answer is “Nothing much”, but sometimes a littlemore can be said.

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ALL ROOTS INTEGRAL

For all kinds of graphical (and other polynomials) a popularquestion is:

What can be said when the polynomial has all roots integral?

Mostly, the answer is “Nothing much”, but sometimes a littlemore can be said.

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CHORDAL GRAPHS

A graph is chordal if it can be constructed from a completegraph 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 integerroots.

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CHORDAL GRAPHS

A graph is chordal if it can be constructed from a completegraph 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 integerroots.

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CHORDAL GRAPHS

A graph is chordal if it can be constructed from a completegraph 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 integerroots.

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CHORDAL GRAPHS

A graph is chordal if it can be constructed from a completegraph 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 integerroots.

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BUT SO DO MANY OTHERS . . .

Many non-chordal graphs have integer chromatic roots.

Hernández and Luca show that finding similarly structuredgraphs with integral chromatic roots is equivalent to findingsolutions to the Prouhet-Tarry-Escott problem.

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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 integerroots 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 thateither b = 0 or a = 1, b = 1

A result of Whitehead saying that a graph co-chromaticwith a 2-tree is a 2-tree

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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 planarchordal graph.

In other words, “the obvious examples are the only examples”.

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DUAL PLANAR CHORDAL GRAPHS

A planar chordal graph has many separating triangles, so adual planar chordal graph has lots of 3-edge cutsets.

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DUAL PLANAR CHORDAL GRAPHS

A planar chordal graph has many separating triangles, so adual planar chordal graph has lots of 3-edge cutsets.

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DUAL PLANAR CHORDAL GRAPHS

A planar chordal graph has many separating triangles, so adual planar chordal graph has lots of 3-edge cutsets.

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PROOF IDEAS

Suppose M = M(G)∗ is a cographic matroid with integralcharacteristic 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 a3-edge cutset in G,Note that flow polynomials “factorize” over 3-edge cutsets.

Apply induction and, as the old Dutch expression goes, “Bert isje oom”!

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STEP 1

If a polynomial

f (z) = zn − a1zn−1 + a2zn−2 − . . .

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

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STEP 1

If a polynomial

f (z) = zn − a1zn−1 + a2zn−2 − . . .

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

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STEP 2

As the flow polynomial is

C(M, z) = zr − |E|zr−1 +

((|E|2

)− γ3

)zr−2 − . . .

an upper bound on ((|E|2

)− γ3

)gives a lower bound on γ3.

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STEP 2

After some slightly fiddly details, and lots of coffee

we conclude that γ3 is strictly larger than the number of verticesof degree 3 in G, and so G has a proper 3-edge cutset.

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FINAL STEP

A flow analogue of the clique cutset formula:

FG(z) =FH(z)FJ(z)

(z− 1)(z− 2)

G

H J

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FINAL STEP

By induction, both H and J are dual planar chordal graphs, andtherefore so is G.

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FINAL REMARKS

A supersolvable matroid is the matroidal analogue of a chordalgraph, and it has integral characteristic roots.

For flow roots, what we really showed was two separate things:

A cographic matroid with integral characteristic roots issupersolvableA supersolvable cographic matroid is the dual of a planargraph

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FINAL QUESTION

QUESTION

Are there other natural classes of (binary) matroids whereintegral characteristic roots implies supersolvability?

Two promising classes to consider:

4-colourable graphs (Dong), andBinary matroids with no M(K5)-minor.

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Thanks for listening!

Hartelijk dank, Bert!

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