Path Ideals for Weighted Graphs
Sean Sather-Wagstaff
North Dakota State University
Date 18 October 2014
AMS Eastern Section Meeting
Dalhousie University
Joint with Bethany Kubik (arXiv:1408.1674)
and Chelsey Paulsen (J. Algebra Appl., 12 (2013), no. 5)
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Edge Ideals
Assumption
k is a field, S = k [x1 , . . . , xd ], and G = (V , E) is a (finite simple)
graph with V = {x1 , . . . , xd }.
Definition (Villareal ’90)
The edge ideal I(G) ⊆ S of G is I(G) = (xi xj ∈ E)S.
Example
G=
xx11
xx22
xx33
xx44
xx55
xx66
I(G) = (x1 x2 , x2 x3 , x1 x4 , x2 x5 , x3 x6 )S.
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Irreducible Decompositions of Edge Ideals
Definition
A vertex cover of G is a subset W ⊆ V such that for every
xi xj ∈ E, either xi ∈ W or xj ∈ W .
Fact
We have (irredundant) irreducible decompositions
\
\
I(G) = (W )S =
(W )S.
W
W min
Example
G=
x1
x2
x3
x4
x5
x6
I(G) = (x1 , x2 , x3 )S ∩ (x1 , x2 , x6 )S ∩ (x1 , x3 , x5 )S
∩(x2 , x3 , x4 )S ∩ (x2 , x4 , x6 )S
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Cohen-Macaulayness of Trees
Theorem (Villareal 1990)
If T is a tree, then S/I(T ) is Cohen-Macaulay if and only if I(T )
is unmixed, if and only if T is a suspension of a tree. (Hence,
Cohen-Macaulayness of S/I(T ) is characteristic-independent.)
Example
T =
x1
x2
x3
x4
x5
x6
S/I(T ) is Cohen-Macaulay.
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Weighted Edge Ideals
Definition
A weight function on G is a function ω : E → N.
A weighted graph Gω is a graph G, with a weight function ω.
Definition (Paulsen-SW ’13)
The weighted edge ideal I(Gω ) ⊆ S of a weighted graph Gω is
ω(e) ω(e)
xj
I(Gω ) = (xi
| e = xi xj ∈ E)S.
Example
Gω =
xx11
22
xx44
33
xx22
44
xx55
11
xx33
55
xx66
I(Gωω) = (x1133x2233, x2 x3 , x12 x42 , x24 x54 , x35 x65 )S.
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Irreducible Decompositions of Weighted Edge Ideals
Definition
A weighted vertex cover W σ of Gω is a vertex cover W ⊆ V with
a function σ : W → N such that for every e = xi xj ∈ E, one has
1
xi ∈ W and σ(xi ) ≤ ω(e), or
2
xj ∈ W and σ(xj ) ≤ ω(e).
σ(xi )
Set (W σ )S = (xi
| xi ∈ W )S.
Example
Gω =
x12
2
x4
3
x2
1
x31
4
5
x54
x6 .
This weighted vertex cover is minimal: no vertices can be
unboxed, and no weights (exponents) can be increased.
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Decompositions of Weighted Edge Ideals
Theorem (Paulsen-SW ’13)
We have (irredundant) irreducible decompositions
\
\
I(Gω ) =
(W σ )S =
(W σ )S
W σ min
Wσ
Example
Gω =
xx121
22
xx44
3
x2
11
xx331
4
4
55
xx554
xx6 .
6
I(Gω ) =(x12 , x2 , x35 )S ∩ (x12 , x24 , x3 )S ∩ (x12 , x2 , x65 )S
∩ (x12 , x3 , x54 )S ∩ (x2 , x35 , x42 )S ∩ (x23 , x3 , x42 )S
∩ (x2 , x42 , x65 )S ∩ (x13 , x24 , x3 , x42 )S ∩ (x13 , x3 , x42 , x54 )S
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Cohen-Macaulay Weighted Trees
Theorem (Paulsen-SW ’13)
If Tω is a weighted tree, then S/I(Tω ) is Cohen-Macaulay if and
only if I(Tω ) is unmixed, if and only if T is a suspension of a
tree Γ such that for each edge vi vj in Γ one has
ω(vi vj ) ≤ min{ω(vi wi ), ω(vj wj )}.
Example
non-CM
3
v1
2
w1
CM
3
v1
3
w1
Sean Sather-Wagstaff
v2
1
v3
4
5
w2
w3
v2
1
v3
4
5
w2
w3
Path Ideals for Weighted Graphs
Weighted Path Ideals
Definition (Kubik-SW)
Fix an integer r ≥ 1. The weighted r -path ideal Ir (Gω ) ⊆ S of a
weighted graph Gω is the ideal of S generated by all monomials
max(ω(xir −2 xir −1 ),ω(xir −1 xir )) ω(xir −1 xir )
ω(xi0 xi1 ) max(ω(xi0 xi1 ),ω(xi1 xi2 ))
xi1
· · · xir −1
xir
xi0
such that xi0 xi1 · · · xir −1 xir is a path in G.
Example
1
If r = 1, then this is I(Gω ).
2
If ω = 1 and G is a tree, then this is Conca’s Ir (G),
generated by all the r -paths in G.
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Weighted Path Ideals
Example
xx11
Gω =
22
xx44
33
xx22
11
44
xx55
xx33
55
xx66
I2 (Gω ) = (x13 x23 x3 , x13 x24 x54 , x2 x35 x65 , x13 x23 x42 , x24 x3 x54 )S.
Theorem (Kubik-SW)
We have (irredundant) irreducible decompositions
\
\
Ir (Gω ) =
(W σ )S =
(W σ )S
Wσ
Sean Sather-Wagstaff
W σ min
Path Ideals for Weighted Graphs
Cohen-Macaulay Weighted Trees
Theorem (Kubik-SW)
Let Gω be a weighted tree with no r -pathless leaves. TFAE:
(i) S/Ir (Gω ) is Cohen-Macaulay;
(ii) Ir (Gω ) is unmixed; and
(iii) Gω is an r -path suspension of a weighted tree Γµ s.t. for all
vi vj ∈ E(Γµ ) one has ω(vi vj ) ≤ min{ω(vi yi,1 ), ω(vj yj,1 )}.
Example
I2 (Gω )
v1
3
v2
1
v3
4
5
y1,1
y2,1
y3,1
a
b
c
y1,2
y2,2
y3,2
3
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs
Cohen-Macaulay Weighted Cliques
Theorem (Paulsen-SW ’13)
Let Kωd be a weighted d-clique. Then S/I(Kωd ) is
Cohen-Macaulay of dimension 1.
Theorem (Kubik-SW)
Let Kω3 be a weighted 3-clique. Then S/I2 (Kω3 ) is CohenMacaulay if and only if Kω3 has edge weights of the form
a = a ≤ b.
Theorem (Kubik-SW)
Let Kωd be a weighted d-clique. Then S/I2 (Kωd ) is CohenMacaulay if and only if S/I2 (Kλ3 ) is Cohen-Macaulay for each
induced weighted sub-clique Kλ3 .
Sean Sather-Wagstaff
Path Ideals for Weighted Graphs