C HARACTERISTIC P OLYNOMIALS WITH
I NTEGER ROOTS
Gordon Royle
School of Mathematics & Statistics
University of Western Australia
Bert’s Matroid Jamboree
Maastricht 2012
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AUSTRALIA
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P ERTH
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W HERE EVERY PROSPECT PLEASES . . .
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. . . AND ONLY MAN IS VILE
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C HARACTERISTIC P OLYNOMIAL
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.
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C HARACTERISTIC P OLYNOMIAL
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.
If M = M(G) is graphic, then C(M(G), z) = z−c PG (z) where
PG (z) is the well-known chromatic polynomial of G.
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C HARACTERISTIC P OLYNOMIAL
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.
If M = M(G) is graphic, then C(M(G), z) = z−c 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.
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E XAMPLES
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
0
1
1
2
3
4
5
6
7
21
7
28
35
21
7
1
7
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 polynomial
is 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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C HROMATIC 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.
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R ESULTS AND C ONJECTURES
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.
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U PPER 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 Kk minor.
1
Proved for graphs by Woodall in 1992, and for general matroids 15 years
earlier by Oxley
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U PPER ROOT- 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 . . .
2
that is, most-wanted by me
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A LL 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?
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A LL 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.
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C HORDAL 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)
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C HORDAL 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)
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C HORDAL 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)
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C HORDAL 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.
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B UT 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.
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P LANAR GRAPHS
However, if we restrict to planar graphs then all is well:
T HEOREM (D ONG & 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
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F LOW ROOTS
Joe Kung and I investigated graphs with integral flow roots.
T HEOREM (K UNG & 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”.
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D UAL PLANAR CHORDAL GRAPHS
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D UAL PLANAR CHORDAL GRAPHS
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D UAL PLANAR CHORDAL GRAPHS
A planar chordal graph has many separating triangles, so a
dual planar chordal graph has lots of 3-edge cutsets.
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P ROOF I DEAS
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”!
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S TEP 1
If a polynomial
f (z) = zn − a1 zn−1 + a2 zn−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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S TEP 1
If a polynomial
f (z) = zn − a1 zn−1 + a2 zn−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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S TEP 2
As the flow polynomial is
r
C(M, z) = z − |E|z
r−1
+
|E|
− γ3 zr−2 − . . .
2
an upper bound on
|E|
− γ3
2
gives a lower bound on γ3 .
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S TEP 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.
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F INAL STEP
A flow analogue of the clique cutset formula:
FG (z) =
FH (z)FJ (z)
(z − 1)(z − 2)
G
J
H
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F INAL S TEP
By induction, both H and J are dual planar chordal graphs, and
therefore so is G.
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F INAL 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
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F INAL QUESTION
Q UESTION
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.
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Thanks for listening!
Hartelijk dank, Bert!
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