Graph Invariants with Connections to the Feynman Period in \phi^{4} Theory.

# Graph Invariants with Connections to the Feynman Period in ϕ4 Theory.

Iain Crump
April 7, 2017
###### Abstract

Feynman diagrams in scalar theory have as their underlying structure -regular graphs. In particular, any -point graph can be uniquely derived from a -regular graph by deleting a single vertex. The Feynman integral is encoded by the structure of the graph and is used to determine amplitudes of particle interactions. The Feynman period is a simplified version of the Feynman integral. The period is of special interest, as it maintains much of the important number theoretic information from the Feynman integral. It is also of structural interest, as it is known to be preserved by a number of graph theoretic operations. In particular, the vertex deleted in constructing the -point graph does not affect the Feynman period, and it is invariant under planar duality and the Schnetz twist, an operation that redirects edges incident to a -vertex cut. Further, a -regular graph may be produced by identifying triangles in two -regular graphs and then deleting these edges. The Feynman period of this graph with a vertex deleted is equal to the product of the Feynman periods of the two smaller graphs with one vertex deleted each. These operations currently explain all known instances of non-isomorphic -point graphs with equal periods.

With this in mind, other graph invariants that are preserved by these operations for -point graphs are of interest, as they may provide insight into the Feynman period and potentially the integral. In this thesis the extended graph permanent is introduced; an infinite sequence of residues from prime order finite fields. It is shown that this sequence is preserved by these three operations, and has a product property. Additionally, computational techniques will be established, and an alternate interpretation will be presented as the point count of a novel graph polynomial.

Further, the previously existing invariant and Hepp bound are examined, two graph invariants that are conjectured to be preserved by these graph operations. A combinatorial approach to the invariant is introduced.

Feynman period, graph invariant, invariant, Hepp bound, matrix permanent, extended graph permanent.
\thesistype

Dissertation \previousdegreesM.Sc., Simon Fraser University, 2012
B.Sc., University of Winnipeg, 2009 \degreeDoctor of Philosophy \disciplineMathematics \departmentDepartment of Mathematics \facultyFaculty of Science \copyrightyear2017 \semesterSpring 2017 \committee\chairDr. Cedric ChauveProfessor \memberDr. Karen YeatsSenior Supervisor
Associate Professor \memberDr. Matt DeVosSupervisor
Associate Professor \memberDr. Bojan MoharInternal/External Examiner
Professor \memberDr. Robert ŠámalExternal Examiner
Associate Professor
Computer Science Institute
Charles University

\makecommittee
###### Acknowledgements.
I would like to thank Francis Brown, Matt DeVos, Erik Panzer, Scott Sitar, and of course Karen Yeats for their helpful notes and ideas, and fantastic support.

List of Figures

\mainmatter

## Chapter \thechapter Introduction

### 1 Background

A goal of a quantum field theory is to understand the various interactions of the fundamental particles of which the universe is composed (see, for example, [13], [23], and [28]). Each such theory restricts its allowable particle types and immediate interactions between these fundamental particles based on experimental observations. These interactions are encoded in Feynman diagrams, fundamentally these are multigraphs with specific edge types, representing the different types of particles. Further, these diagrams allow external edges, edges that connect to one vertex only. These external edges represent particles as they enter or exit the system. We may therefore view edges in the diagram as pairings of half-edges, while external edges are half-edges that remain unpaired.

The following example uses a framework that can be found in [52].

###### Example.

Quantum electrodynamics is the quantum theory of electromagnetism. It has three half-edge types; a front-half fermion, a back-half fermion, and a photon. Edges may be constructed by combining front- and back-half fermions to create a fermion edge, , or by combining two photon half-edges, . There is one interaction type, consisting of one of each half-edge types. A fermion edge oriented in the direction one is reading represents an electron, and against represents a positron. Reading Figure 1 left to right, then, an electron and positron combining to form a photon, which propagates for a time before splitting into an electron and positron again.

This thesis will focus on theory111Here, is the usual name of the field. The field appears to the forth power in the Lagrangian, giving the -valent vertices and also the name ., which has allowable half-edge , an undecorated and undirected edge, and interaction . In particular, a -point graph has precisely external edges, and when considering the motivating physics we restrict to -point graphs in theory. These graphs can be uniquely derived from -regular graphs by deleting a single vertex and all incident half-edges, leaving the remaining half-edges unpaired.

The Feynman diagram encodes the Feynman integral. Let be the dimension of space-time appropriate to the physical theory. Both quantum electrodynamics and theory, for example, are -dimensional physical theories, corresponding the standard three space and one time dimensions. The loop number (also known as the first Betti number) of a Feynman diagram , , is the minimum number of edges that must be removed to produce an acyclic graph. Suppose the Feynman diagram has internal edges, external edges, and each internal edge is assigned an orientation and has momentum and mass . We impose momentum conservation at each vertex; the sum of momenta in must be equal to the sum of momenta out. This is akin to the graph theoretic notion of flows that will arise in numerous places throughout this thesis. As a result, we may express momenta flowing through internal lines as a linear combination of the independent loop momenta and the external momenta as

where the and values are determined by the direction of these internal lines and the orientation of these edges. From Equation 1.11 in [48], the scalar Feynman integral as a potentially divergent formal integral expression is

 ∫(RD)hGhG∏r=1dDkrn∏j=11(q2j−m2j).

The first step to make a convergent integral is to regularize. Two common choices are to raise each factor in the denominator to a non-integer parameter, or to take as (see [7]). Multiplication of vectors in is taken to be the dot product, following standard physics notations. Non-scalar field theories result in more complicated Feynman integrals, as every edge and vertex will be represented by more complicated factors in the integral. However, theory is a scalar field theory, so this representation is sufficient for our needs.

After parametrization222In literature, this is known as Schwinger parametrization or Feynman parametrization. the Feynman integral is (Equation 1.12 in [48])

 ∫xi≥0UhG(D/2)−nGΨ(hG+1)(D/2)−nG∣∣ ∣∣xn=1n−1∏i=1dxi

(see [39], Chapter 2, for a detailed explanation of how the first integral is translated to this form). In the denominator, is the Kirchhoff polynomial (also known as the first Symanzik polynomial). Let be the set of spanning trees of diagram , and for all assign variable , the Schwinger parameter. The Kirchhoff polynomial of is

 ΨG=∑T∈TG⎛⎝∏e∉Txe⎞⎠.

This was introduced in [30] as a tool for understanding electrical systems. Similarly, is the second Symanzik polynomial. The sum in the second Symanzik polynomial is over , the set of spanning forests with precisely two connected components. Using conventions from [7], and writing as the set of external momenta in tree ,

 UG=∑(T1,T2)∈FG⎛⎝∏e∉(T1∪T2)xe⎞⎠⎛⎜⎝∑pj∈PT1∑pk∈PT2pjpk⎞⎟⎠+ΨGn∑i=1xim2i.

It is of interest that the Feynman integral tends to diverge. Renormalization, given mathematical foundation in [20], is the method by which this integral, as a part of a larger computation, is made to match experimentally observed values. A brief historical review can be found in [26].

### 2 Feynman periods

#### 2.1 Derivation from the Feynman integral

Our interests lie in the Feynman period. For a diagram with internal edges, the Feynman period is

 ∫xi≥01ΨD/2|xn=1n−1∏i=1dxi,

a residue of the Feynman integral of viewed as a Feynman diagram in massless scalar field theory, with all external parameters and masses set to zero. As a result, we may ignore completely all external edges, and the Feynman diagrams truly are graphs or multigraphs. Throughout, standard graph theory terminology and notation will be assumed, following [8]. The period is an important object (see, for example, [6, 12, 33]). When the Feynman integral diverges, the period itself is the coefficient at infinity (see Section 6.2.3 in [28]). When the period converges, it does so independent of renormalization scheme. It is also understood precisely when the period is convergent, which we describe now.

Recall the loop number of a graph , , is the minimum number of edges that must be removed to produce an acyclic graph. A graph is primitive (often primitive log divergent in the literature) if and for all proper subgraphs , . Any subgraph that defies this relation is called a subdivergence. For a -regular graph, a subdivergence is a non-trivial - or -edge cut.

###### Theorem (Proposition 5.2 in [6]).

The period of a graph converges if and only if the graph is primitive.

Despite being a simplification of the Feynman integral, the period is still difficult to compute. Many can be expressed as multiple zeta values; sums and products of

 ζ(s1,...,sk)=∑n1>⋯>nk>01ns11⋯nskk

over for positive integers , . These are objects of mathematical interest, due to the algebraic and number theoretic properties they possess, and their connections to other mathematical objects (see for example [27] and [54]). Computations for -point graphs up to seven loops are presented by Broadhurst and Kreimer in [11], up to eight loops by Schnetz in [43], and for an important and infinite family of graphs by Brown and Schnetz in [16]. Both [11] and [43] use numeric techniques to produce numbers that match with consistently high degrees of accuracy, while [16] relies on recently developed tools to establish the equality with mathematical rigor. One way to deal with this computational difficulty of the integral Feynman period is to create large families of graphs with equal periods. To do this, we rely on graph operations that preserve the period. Four such operations are known for primitive -point graphs.

#### 2.2 Operations that preserve the Feynman period

As observed in Section 1, any -point graph can be uniquely created from a -regular graph by deleting a vertex. This operation is called decompletion, and the unique way a vertex can be added back to a -point graph is completion.

###### Theorem (Theorem 2.7 in [43]).

The period of a -point graph is invariant under completion followed by decompletion.

As the graphs in the set of decompletions of a -regular graph are not necessarily isomorphic, this can produce a number of non-isomorphic graphs with equal periods.

###### Example.

Consider the graphs drawn in Figure 2. Graph is -regular, while and are decompletions of . As and have different numbers of edges in parallel, they are clearly non-isomorphic.

The Schnetz twist is another operation known to preserve the period. Shown in Figure 3, we partition the edges of a completed graph across a four-vertex cut. On one side of this cut, we then change the ends of edges of the form , exchanging with and with , using the labeling from this figure. We assume that both graphs are -regular.

###### Theorem (Theorem 2.11 in [43]).

If two -regular graphs differ by a Schnetz twist, then any pair of decompletions of these graphs have equal periods.

###### Example.

The completed graphs in Figure 4 use the naming convention from [43]. There is a -vertex cut, aligned vertically in the centre, and the two graphs differ by a Schnetz twist, performed on the edges to the right of the cut. These are non-isomorphic; contains five triangles, contains six.

Finally, the dual of a connected planar graph is a well-known graphic operation. Importantly, a -point graph has . Recall Euler’s polyhedral formula; setting as the set of faces for an embedding of a connected graph , . If is planar, then dual has . Then, , and hence . As also, both and have the same vertex to edge relationship.

###### Theorem (Theorem 2.13 in [43], used as early as [11]).

Suppose graph and dual are -point graphs. The periods of and are equal.

###### Example.

Again using the naming convention from [43], the graphs in Figure 5 are decompletions of and , respectively. Both are -point graphs. These are non-isomorphic; the vertices of degree three in are an independent set, while those in are not.

Lastly, while not period preserving itself, decompleted graphs with -vertex cuts have an important property with regards to the period. Split the graph into minors and as in Figure 6 and assume that , , and are all -point graphs. External half-edges must be distributed as shown. The operation is shown for completed graphs in Figure 7.

###### Theorem (Theorem 2.10 in [43]).

Suppose graphs , , and in Figure 6 are all -point graphs. The period of is equal to the product of the periods of and .

It is therefore possible to produce non-isomorphic -point graphs with equal periods by merging smaller graphs in this manner; either merging the same two graphs at different pairs of edges, or merging different graphs that differ by an aforementioned graph operation. From a collection of graphs with known periods, it is possible to produce an infinite family of graphs with easily computed periods, unlike the previous structural operations.

###### Example.

In Figure 8, the graph is a decompletion of and is a decompletion of . Two ways of merging these graphs are shown below. These are again non-isomorphic, as the number of triangles is different between the two graphs.

The four operations listed above explain all currently known instances of -point graphs with equal periods. Any non-trivial graph invariant that is preserved by these is therefore of interest, as it may provide insight into the Feynman period. The creation of such an invariant is one of the main topics of this thesis.

### 3 Outline of the thesis

Chapters Graph Invariants with Connections to the Feynman Period in Theory. and Graph Invariants with Connections to the Feynman Period in Theory. introduce the Hepp bound and invariant, respectively. These are non-trivial invariants conjectured to be preserved by all operations known to preserve the period. A novel graph-theoretic interpretation of the invariant is introduced in Section 5.

Chapter Graph Invariants with Connections to the Feynman Period in Theory. provides a brief introduction to matroid theory, and some of the tools therein that will be of use. Those familiar with the subject may safely skip this chapter.

We previously created the extended graph permanent, and introduce it here in Chapter Graph Invariants with Connections to the Feynman Period in Theory.. For an arbitrary graph, the extended graph permanent is a sequence of residues from (an infinite subset of) prime order finite fields. Constructed from the matrix permanent, Section 6 lays the foundation and establishes the required linear algebra tools, and Section 7 introduces the invariant. Useful to our understanding of the extended graph permanent as a combinatorial object, we introduce a graphic formulation of this object in Section 8. Section 9 briefly discusses a natural sign ambiguity that exists in this invariant. Section 10 explores a connection between the extended graph permanent and nowhere-zero flows.

In Chapter Graph Invariants with Connections to the Feynman Period in Theory. we prove that the extended graph permanent is in fact preserved by all previously mentioned operations known to preserve the period.

###### Theorem.

Let be a -regular graph.

• For , the extended graph permanent of is equal to the extended graph permanent of (Theorem 48).

• If and are both -regular graphs that differ by a Schnetz twist, then any decompletions of and will have equal extended graph permanents (Proposition 49).

• For , let . If is planar and is its planar dual, then and have equal extended graph permanents (Corollary 65).

• If has a -vertex cut that can be split into two -point minors as in Figure 6, then the extended graph permanent of is the additive inverse of the (term-by-term) product of the extended graph permanents of and (Theorem 68).

This of course leads naturally to the following conjecture.

###### Conjecture (Conjecture 4).

If two -point graphs have equal periods, then they have equal extended graph permanents.

This connection to the Feynman period is further hinted at in Theorem 70, which proves a similar term-by-term product property for -regular graphs with -edge cuts, corresponding to graphs with subdivergences.

Chapter Graph Invariants with Connections to the Feynman Period in Theory. develops computational methods for finding the extended graph permanent. Specifically, permanent values are difficult to compute, and the extended graph permanent produces sequences that, even for small graphs, quickly require residues modulo prime of the permanents of obscenely large matrices. In this chapter, we develop methods for finding closed forms for all values in the sequence for any graph. We further find closed forms for the extended graph permanent for the family of trees in Section 17, wheels in Section 18, and zig-zag graphs in section 19. In producing these sequences we also observe a possible connection to the invariant.

In Chapter Graph Invariants with Connections to the Feynman Period in Theory. we find a novel graph polynomial such that the extended graph permanent can be represented as a point count over this polynomial. For function and prime , let be the number of roots of over finite field .

###### Theorem (Corollary 85).

Let be a -point graph. The extended graph permanent of at is

 {[˜FG]p(modp) if |E(G)|≡0(mod4)−[˜FG]p(modp) otherwise.

A number of extended graph permanent sequences also appear to relate to modular forms, as the Fourier coefficient modulo . This is discussed in greater detail in Section 23. It is interesting to note that in all observed instances, the loop number of the graph is equal to the weight of the modular form, the level of the modular form is a power of two, and in the Dirichlet character decomposition these all fall into subspaces of dimension .

Finally, we conclude with Chapter Graph Invariants with Connections to the Feynman Period in Theory.. Here, we summarize the main results of this thesis, and indicate areas of potential future interest.

## Chapter \thechapter The Hepp bound

The Hepp bound was introduced by Panzer in 2016. The properties and results discussed here are due to Panzer unless otherwise stated ([40]). New material on the Hepp bound has also been published in [42].

For a graph , the Hepp bound is an upper bound for the period, created by replacing the Kirchhoff polynomial in the period formula with the maximal monomial at all points of integration;

 H(G)=∫xi≥0∏|E(G)|−1i=1dxi(maxT∈TG{∏e∉Txe})2|x|E(G)|=1∈Q.

By integrating over smaller denominators, this naturally creates an upper bound for the Feynman period.

###### Example.

The banana graph, using the naming convention from [43], is the unique graph with two vertices, and two edges in parallel between them. The Kirchhoff polynomial for this graph is . The Hepp bound of this graph is therefore

 H(P1,1)=∫x1≥0dx1max{1,x1}2 =∫1x1=0dx11+∫∞x1=1dx1x21 =1+(−x1)−1|∞1 =1+(0−(−1))=2.

From [41], the period of this graph is .

Computational complexity rises quickly. The next smallest primitive -point graph, , has sixteen summands in its Kirchhoff polynomial. A more graphic and structural interpretation of the Hepp bound makes this computation easier. To do this, we define to be the set of maximal chains of bridgeless subgraphs of ;

 F(G)={∅=γ0≠γ1⊊γ2⊊⋯⊊γh1(G)=G: each γi is% bridgeless}.

The length of these chains and the fact that follows from the required maximality.

###### Proposition 1 ([40]).

Let be a graph and the set of maximal bridgeless chains. Define . Then,

 H(G)=∑[γ]∈F(G)1ω(γ1)⋯ω(γh1(G)−1)h1(G)∏k=1(|E(γk)|−|E(γk−1)|).

From a graph theoretic standpoint, these chains resemble ear decompositions (see Section 5.3 in [9]). The key distinction is that the subgraphs in the chains may be disconnected. Any subgraph that has fewer connected components than its predecessor must add two edge-disjoint paths between these connected components.

###### Example.

Consider a decompletion of graph . There are, up to symmetries, two possible maximal chains of bridgeless subgraphs;

 \raisebox−0.4pt\includegraphics[]hepp0⊊\raisebox−0.4pt\includegraphics[]hepp1⊊\raisebox−0.4pt\includegraphics[]hepp2⊊\raisebox−0.4pt\includegraphics[]heppfull

and

 \raisebox−0.4pt\includegraphics[]hepp0⊊\raisebox−0.4pt\includegraphics[]hepp3⊊\raisebox−0.4pt\includegraphics[]hepp4⊊\raisebox−0.4pt\includegraphics[]heppfull.

There are twelve chains of the first type, contributing to the sum, and six chains of the second type, contributing to the sum. Using Proposition 1, it follows that . Broadhurst proved in [10] that the period of this graph is .

As an upper bound, the Hepp bound does not appear to be incredibly precise. However, plotting the periods of primitive -point graphs against these Hepp bounds for graphs up to eleven loops, there was an exceptionally strong correlation at every loop number. Panzer observed that the period of a graph was approximately , with error approximately one percent. The data thus inspires the following conjecture.

###### Conjecture 1 ([40], Conjecture 3.2 in [42]).

For primitive graphs, the Hepp bounds are equal if and only if the periods are equal.

This is unique among invariants studied here. For both the invariant (Conjecture 2 in Chapter Graph Invariants with Connections to the Feynman Period in Theory.) and the extended graph permanent (Conjecture 4 in Chapter Graph Invariants with Connections to the Feynman Period in Theory.), it is conjectured only that if two graphs have equal period, then they have equal invariant and extended graph permanent. In both cases, the converses of these statements are known to be false.

Of course, Conjecture 1 implies that the Hepp bound must be preserved by completion followed by decompletion, planar duality, and the Schnetz twist. Further, joining two graphs at a pair of edges as in Figure 6 should result in some sort of product property. Two of these properties have been established.

###### Proposition 2 (Panzer).

If the graph is planar, and its planar dual, are both -point graphs in theory, then and have equal Hepp bounds.

###### Proposition 3 ([40]).

Suppose the graph is made from joining graphs and , as in Figure 6. If these graphs are all -point ,

 H(G)2=H(G1)2H(G2)2.

All other invariance properties remain conjectural.

Conjecture 1, and additionally the approximation given prior, suggest that there is a surprising usefulness to the Hepp bound. The fact that the upper bound lacks precision but scales fantastically to a very precise estimate indicates the value of future research. In particular, the Hepp bound is a considerably more easily computed value than the period itself; the Hepp bound can be computed in exponential time, while much of the difficulty in computing the period arises from the absence of an algorithm. In this regard, a number of graphs have known period and Hepp bounds equal to graphs with periods that have not been resolved. It follows immediately from Conjecture 1 that the periods of and are conjectured to be equal, and that periods of and are conjectured to be equal.

## Chapter \thechapter The c2 Invariant

### 4 The c2 invariant and a conjectured connection to the Feynman period

The was first introduced in [44] and further developed in [14]. Recall from the introduction that the Kirchhoff polynomial for a graph is

 ΨG=∑T∈TG⎛⎝∏e∉Txe⎞⎠∈F[x1,...,x|E(G)|],

where is the Schwinger parameter for each edge , is the set of spanning trees of , and can be any field. We define the point count of a function over finite field , denoted , to be the number of roots of in . Let be the increasing sequence of all prime integers. Let be a graph such that . It can be shown that is divisible by (see [44]). We define the invariant for at prime as , and the invariant of as the sequence of residues .

###### Example.

Consider the graph . As any two edges form a spanning tree, . For prime , any values will force a unique value for as a solution to in . Hence, , and the invariant for is equal to for all primes.

It is known that the Kirchhoff polynomial for a graph can also be represented as the determinant of a matrix. Applying an arbitrary orientation to the edges of , let be a signed incidence matrix of , where rows are indexed by vertices and columns are indexed by edges;

 mv,e=⎧⎨⎩1, if h(e)=v−1, if t(e)=v0, otherwise,

where is the head of edge and is the tail. Some authors use opposite signs for entries, but this is ultimately arbitrary in the construction. We create the matrix from by deleting a row associated to an arbitrary vertex; call this the reduced signed incidence matrix. Let be the diagonal matrix with entries for , edges ordered to align with . Define the modified Laplacian matrix to be

 KG=[AMTG−MG0].
###### Proposition 4.

With notation as defined prior, , regardless of choice of orientation or row deleted in constructing .

The following lemma was first presented by Kirchhoff in [30]. The proof of Proposition 4 that follows is from [12].

###### Lemma 5 ([30]).

Let be a graph and such that , the loop number of . Let denote the square matrix obtained from the previously defined reduced signed incidence matrix by deleting the columns of indexed by elements of . Then,

 det(MG(I))={±1 if I is a spanning tree of G0 otherwise.

We further require, for the proof of Proposition 4, the Leibniz formula for the determinant of an matrix ,

 det(A)=∑σ∈Snsgn(σ)n∏i=1ai,σ(i),

where the sum is over all elements of the symmetric group , and is the signature of .

###### Proof of Proposition 4.

From the shape of the modified Laplacian matrix , and using the Leibniz formula for the determinant,

The restriction in the summation comes from the fact that if , the columns of the matrix are not independent, and therefore the determinant vanishes. From Lemma 5, if is a spanning tree and zero otherwise. Therefore,

###### Example.

One possible reduced signed incidence matrix for the graph is

 MK3=[110−101].

A modified Laplacian for this graph is therefore

 KK3=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣x1001−10x201000x301−1−100010−100⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

The determinant of this matrix is , which agrees with the Kirchhoff polynomial.

As stated in the introduction, it is believed that the invariant is preserved by all operations known to preserve the period. Conjecture 2 then follows.

###### Conjecture 2 (Conjecture 5 in [14]).

If two -point graphs have equal periods, then they have equal invariants.

It remains an open problem that the invariant is preserved by completion followed by decompletion and the Schnetz twist. Duality is established in [25].

###### Theorem 6 (Theorem 39 in [25]).

If planar graph and its dual are both -point graphs, then .

Lastly, -point graphs with two-vertex cuts do not have a product property, but two graphs constructed using this method will have equal invariants.

###### Proposition 7 (Proposition 16 in [15]).

Let be a graph with a -vertex cut. Suppose we may split the graph into two minors and across this cut, as in Figure 6. If , , and are all -point graphs, then for all primes .

Similarly, subdivergences also result in the invariant vanishing.

###### Theorem 8 (Theorem 38 in [17]).

If a graph contains a subdivergence, then for all primes .

### 5 Graphic interpretation of the c2 invariant

In my attempts to work with the invariant, especially with the hope of proving that two graphs that differ by completion followed by decompletion had equal invariants, I believed it would be useful to work with a more structural interpretation of the invariant. In this section we construct this interpretation. This is original work.

Throughout this section, assume that is prime. Let be a connected graph. By Proposition 4, we may consider a zero of the Kirchhoff polynomial of over as an assignment of values to the Schwinger parameters such that the modified Laplacian matrix has determinant zero. Hence, this is an assignment of values to the Schwinger parameters such that there is a nontrivial linear combination of the rows of that sums to the zero vector over . Fix such a linear combination, and let be the coefficient given to the row indexed by edge . Since is connected and a row of was deleted in the construction of , the rows of are linearly independent, and hence at least one row in the first must receive a nonzero coefficient in this linear combination. Whence, we may consider these coefficients on the first rows as weights on the edges of . Specifically, consider the signed incidence matrix in the upper right block. Denoting the head and tail of edge and and , respectively,

 ∑e∈E(G)h(e)=vwe−∑e∈E(G)t(e)=vwe=0

for all vertices .

###### Definition 9.

Let be a directed graph and an abelian group. Let be a weighting of the edges. We define the associated boundary function by

 (∂v)ϕ=∑e∈E(G)h(e)=vϕ(e)−∑e∈E(G)t(e)=vϕ(e).

We say that is a flow if, for all , in .

Hence, a linear combination of the matrix rows that sums to the zero vector can be thought of as assigning weights to the edges of the associated graph that creates a non-trivial flow. As stated in the introduction, this is comparable to momentum conservation in Feynman graphs. See [53] for a more general introduction to flows.

###### Example.

Consider the graph , with orientation as drawn below. The modified Laplacian is included with this graph, vertex labels are assigned arbitrarily and edges are ordered lexicographically.

 \raisebox−0.4pt\includegraphics[scale=0.8]c2k4,KK4=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣x1100x2010x3001x4−110x510−1x60−11−1001−100−10−10100−101−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

Considering only the upper right block, we may create a linear combination in that sums to the zero vector as

 433230⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣100010001−11010−10−11⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

Treating these coefficients as weights on the edges, this linear combination does indeed translate to an flow on the graph,

 \raisebox−0.4pt\includegraphics[]c2flow.

Again, fix a linear combination of the rows of a modified Laplacian matrix that sum to the zero vector over . Consider now the coefficients on the last rows in this linear combination. These may similarly be used to describe weights on the vertices; for denote this weight , and again let be the weight on edge . Then, for edge , the Schwinger parameter for the edge must be a value that balances the equation

 wexe+wa−wb=0,

where for the vertex that was deleted in the creation of . Consider traveling around a cycle in the underlying unoriented graph; adding when moving with the underlying orientation of the edges and subtracting when not. This will necessarily sum to zero. As a result of the orientation determining whether each term is added or subtracted, all vertex weights will cancel.

###### Definition 10.

Let be an oriented graph, and an abelian group. Let be a weighting of the edges, and the collection of cycles in the underlying unoriented graph. Let be arbitrary. Traveling around , if the orientation of an edge agrees with the direction of travel we put , and otherwise. The boundary function of cycle given is

 (∂C)τ=∑e∈C+τ(e)−∑e∈C−τ(e).

We say that is a -tension if for all , .

See, for example, [36] and [37].

We see then that the Schwinger parameters are values that, multiplying edge-by-edge, turn a non-trivial flow into a tension. In the other direction, consider an flow and an assignment of Schwinger parameters such that is an tension. We may create coefficients for the rows indexed by vertices uniquely by equation and the fact that we may think of the row removed in creating the matrix as assigning weight zero to that vertex. This then defines the coefficients of a linear combination of the row vectors of this modified Laplacian that sums to the zero vector. Hence, this matrix has determinant zero. Returning to this product on individual edges of flows and Schwinger parameters, a map of Schwinger parameters is a zero in if and only if there exists a non-trivial flow such that is a tension. Note that by this construction, we are able to ignore the vertex weights and may consider only the edges. We will call such maps Schwinger solutions (or Schwinger solutions to flow ).

###### Example.

Consider the graph and flow established in the previous example;

 \raisebox−0.4pt\includegraphics[]c2flow.

We consider creating Schwinger parameters such that for each edge, the product of the Schwinger parameter and the weight of the flow creates a -tension in the graph. Such a collection of Schwinger parameters is given in parentheses below, including the original flow and the product for each edge in . The variable may be any value in , as the edge has weight zero in the flow.

A quick check reveals that this is an tension.

Further, we may check that this variable assignment to Schwinger parameters does create a zero of in . That is,

 ΨK4 =x4x5x6+x1x3x4+x2x3x6+x1x2x5+x1x3x5+x1x3x6+x2x3x5+x2x3x4 +x1x2x4+x1x2x6+x2x4x6+x2x4x5+x1x4x6+x1x5x6+x3x5x6+x3x4x5 =1⋅1⋅x+1⋅2⋅1+2⋅2⋅x+1⋅2⋅1+1⋅2⋅1+1⋅2⋅x+2⋅2⋅1+2⋅2⋅1 +1⋅2⋅1+1⋅2⋅x+2⋅1⋅x+2⋅1⋅1+1⋅1⋅x+1⋅1⋅x+2⋅1⋅x+2⋅1⋅1 =0.

The following is a well-known result regarding the number of flows in a graph.

###### Proposition 11.

A connected graph has flows.

###### Proof.

Let be the edges of a spanning tree of . Assign edges in values from arbitrarily. We will show that edges in can be assigned weights back uniquely.

Any vertex incident to precisely one edge that has not received a value may have that value assigned uniquely to maintain the flow property. Removing edges from as values are assigned, we may give values to the leaves of in this manner until only a single edge has no weight assigned, call it . At this point, two vertices are incident to a single edge with no assigned Schwinger value, and we must check that the desired assignments agree. Summing over edges in with assigned weights, say , we know that , as each weight is added once as an head and subtracted once as a tail. As all other vertices have , , and a weight can be added uniquely to , completing the flow in . ∎

A similar method may be used to count the number of Schwinger solutions for a fixed modular flow. The following lemma and corollary will help us prove this property.

###### Lemma 12.

Let be a graph. The sequence of arcs of a closed walk in can be decomposed into a collection of (possibly trivial) cycles that maintain the direction created by the walk and a set of edges which are traversed an equal number of times in opposite directions.

###### Proof.

For a closed walk on graph , create directed multigraph on the same vertex set by adding a directed edge to for each edge in the closed walk in following the direction of the walk. By construction, every vertex has equal in- and out-degree. Removing a maximal collection of directed cycles containing disjoint edges from must leave nothing but edges in pairs and , which completes the proof. ∎

###### Corollary 13.

Let be a directed graph and an edge weighting for some prime . Define the weight of any walk in the underlying graph as the sum of the weights of edges in the walk that agree with the orientation, minus the sum of weights of edges that do not. If every cycle has weight zero in , then so does every closed walk.

###### Proof.

This follows immediately from Lemma 12, as the decomposition presented therein necessarily sums to zero. ∎

###### Proposition 14.

Fix an flow for a graph . Suppose a connected spanning subgraph that contains all edges of weight zero in the flow must have at least edges. There are Schwinger solutions to this flow.

###### Proof.

Assign values to the Schwinger parameters of these edges arbitrarily. Treat tension weights on edges as the product of the Schwinger parameters and the values assigned by this fixed flow. By construction, all Schwinger parameters that have not received a value must correspond to nonzero weights in the flow. Every edge that has not received a Schwinger parameter value will necessarily create a cycle with the edges that have. As such, its tension weight is unique, and so the Schwinger parameter is unique.

We will prove that it is possible to assign values to the remaining Schwinger parameters to create a tension by induction. The base case – the initial assignment of Schwinger values to the edges – cannot create any cycles that break the tension property, as all cycles must be made entirely of weight zero edges. Suppose then that at some stage of the process the assignment of a values to the Schwinger parameters has created only cycles that have the tension property. Consider the next assignment of a Schwinger parameter value. If only a single new cycle will be completed, the assignment is immediate, so we suppose multiple cycles will be completed. Suppose the edge whose Schwinger parameter is being assigned a value is . Then, these two cycles currently create two -paths in the graph on edges that have received Schwinger parameter values. We may therefore create a closed walk in the graph by walking the edges of one -path followed by the edges of the other traversed in reverse. By Corollary 13, this walk may be decomposed into cycles and edges traversed an equal number of times in opposite directions. By induction then, all cycles sum to zero, and the remaining edges cancel. Hence, this closed walk must also sum to zero. Therefore, these two -paths sum to equal values in , and the Schwinger value can be uniquely assigned to . By induction, this completes the proof. ∎

Similarly, we may count the number of flows that have a particular mapping of the Schwinger parameters as a Schwinger solution.

###### Proposition 15.

Let be a graph. Fix an assignment of values in to the Schwinger parameters. The set of flows for which this assignment forms a Schwinger solution forms a vector space.

###### Proof.

Let and be modular flows and a Schwinger solution to both flows. Let be a constant in . Summing around cycle ,

 (∂C)(ϕ1+ϕ2)=∑e∈C+(ϕ1(e)+ϕ2(e))xe−∑e∈C−(ϕ1(e)+ϕ2(e))xe=(∂C)ϕ1+(∂C)ϕ2=0

and

 (∂C)(cϕ1)=∑e∈C+cϕ1(e)xe−∑e∈C−ϕ1(e)xe=c(∂C)ϕ1=0.

Hence, the set of flows for which a fixed assignment of values to the Schwinger parameters is a Schwinger solution is a vector space. ∎

###### Proposition 16.

Let be a graph with some arbitrary orientation on its edges. Create matrix by assigning numeric values in to variables in the modified Laplacian matrix . The nullspace of uniquely defines the set of flows for which this assignment of edge weights is a Schwinger solution, and the number of such flows is .

###### Proof.

Let be an element of the nullspace, so . We may, as before, think of the first entries of as weights on the edges, this time the result of the block . This must define a flow, this time in the graph made from by reversing the direction of every edge. This weighting must then also give a flow in . To show that these vectors give unique flows, note that if , and the elements in corresponding to columns indexed by and are assigned, then the value corresponding to is uniquely determined. Then, as we may thing of the row deletion in the creation of as assigning that vertex weight , this uniqueness propagates out from this vertex. Hence, if two vectors and are in the nullspace and correspond to the same flow, it must be the case that . The number of flows is therefore immediate from the dimension of the nullspace. ∎

Note that the previous two propositions allow for the trivial flow, which of course would be a solution to all Schwinger values.

###### Proposition 17.

Let be flows for a graph . The intersection of sets of Schwinger solutions of these flows has cardinality a power of , and is a subspace of .

###### Proof.

For each flow and each cycle in , create an equation as follows. For cycle , add equation

 (−1)ce1wj,1x1+(−1)ce2wj,2x2+⋯+(−1)cenwj,nxn=0,

where the if the edge is oriented in the direction the cycle is traversed and otherwise, and are from flow . Simultaneously solving for all cycles as a system of equations, if is the number of free variables, then there are solutions. ∎

The goal with the preceding work was to find a structural interpretation of the invariant, possibly for computational simplicity, though ideally to prove that the invariant is unchanged by completion followed by decompletion. Ultimately, inclusion-exclusion produced no useful results.

We lastly explore some potentially useful results regarding the open problems for the invariant. By Proposition 7 and assuming decompletion invariance per Conjecture 2, if a completed graph has connectivity , it has invariant zero for all primes. Hence, we may restrict our investigation to graphs with -connected completion.

###### Menger’s Theorem (Edge-connectivity version [35]).

A graph is -edge-connected if and only if every pair of distinct vertices is connected by edge-disjoint paths.

By Theorem 11, the fact that the number of flows is completion invariant is trivial. It is interesting that there is a natural bijection between flows for these decompletions that follows from Menger’s Theorem, as we may assume that the graph has no subdivergences, and hence no -edge cuts.

Let and be distinct vertices in the completed graph. By Menger’s Theorem, we may fix four edge-disjoint paths between these vertices. Consider one such path and a flow in . Adding vertex back to the graph and assigning all incident edges weight zero, we then preserve the flow structure of the graph. If the edge on this fixed path incident to has weight , we may then subtract from this edges weight, and all weights on edges on this path oriented along the path in the same direction as , and add it to all other edges. By construction, vertices along this path still have the flow property, and edges incident to may now have nonzero weight. If we do this for all paths, a quick check of the four possible cases of pairs of oriented edges incident to and on these paths reveals that vertex also has the flow property. This therefore creates a flow where all edges incident to have weight zero. As such, this can be turned into a flow in . Clearly, this operation is bijective. Unfortunately, the change in the distribution of cycles and additionally of edges of weight zero in the flow makes this unlikely to be of use in proving decompletion invariance for the invariant.

A similar method will be of use in later work, in particular Theorem 48.

Consider now an oriented planar graph. With the orientation, we may distinguish between the sides of an edge, and hence in creating the planar dual may canonically orient the edges of the dual, for example by demanding all edges in the dual travel from the right to the left of the edge in the original graph. Note that with this method of creating the dual, , as the orientation of all edges is reversed. For our purposes, though, this is not a hindrance.

###### Lemma 18.

Let be an oriented planar graph and suppose . If every facial cycle has , then is a tension.

###### Proof.

As the bounded facial cycles of a planar graph can be used as a basis for the cycle space, the proof of this is similar to that of Proposition 14; an arbitrary cycle can be represented as a binary sum of facial cycles. Traversing all of these facial cycles clockwise, then, this sum is zero, as all facial cycles have the tension property, and all edges in a facial cycle but not in are traversed twice, but in opposite directions, and hence cancel. ∎

###### Theorem 19.

Let be a planar graph, and fix a planar orientation. Suppose is an flow. Preserving edge weights from and using the above canonical orientation of the dual, defines an tension in .

###### Proof.

By construction, a facial cycle in the dual comes from the edges incident to a single vertex in the original graph. Therefore, we may travel around this cycle in a direction such that and Immediately, this cycle has the tension property. By Lemma 18 then, this is a tension. ∎

Another graph theoretic method for examining the invariant will be discussed later. Additional tools required to make use of this will be introduced in subsequent chapters, and this alternate method will be introduced in Chapter Graph Invariants with Connections to the Feynman Period in Theory..

## Chapter \thechapter A brief introduction to matroid theory

As some matroid tools will be of use, we include here a short introduction to matroid theory. Notational conventions are taken from [38].

First presented in [50], a matroid is a pair such that is a finite set and is a collection of subsets of with the following properties:

• ,

• if and , then , and

• if and , then there is an element such that .

We call the ground set. An element of is said to be independent, and any other subset of is dependent. A minimally dependent set is a circuit. We may provide an alternate axiom set based on the collection of circuits, . For ground set and collection of subset of , is the set of circuits of a matroid on if and only if (Corollary 1.1.5 in [38]);

• ,

• if and , then , and

• if , , , then there exists a such that .

From these, we may present two fundamental classes of matroids, constructed from matrices and graphs.

###### Proposition 20 (Proposition 1.1.1 in [38]).

Let be the set of columns of an matrix over a field , and the set of subsets of that are linearly independent in the vector space . Then, is a matroid (the vector matroid).

###### Proposition 21 (Proposition 1.1.7 in [38]).

Let be a graph, the set of edges of , and the collection of edge sets of cycles of . Then is the set of circuits of a matroid on (the cycle matroid of ).

It is of particular interest that the cycle matroid of a graph can be represented as a vector matroid over any field using the signed incidence matrix seen in Chapter Graph Invariants with Connections to the Feynman Period in Theory.. A matroid that can be represented as a vector matroid over every field is a regular matroid. Seymour proved in [45] that every regular matroid can be constructed from graphic matroids (matroids that may be represented as the cycle matroid of a graph), cographic matroids (the matroidal dual of graphic matroids, to be introduced shortly), and the matroid (which will be introduced in Section 20), using a particular piecing operation.

Another alternate description of a matroid comes from bases, maximal independent sets. Let be a set and a set of subsets of . Then is a collection of bases of a matroid on if and only if (Corollary 1.2.5 in [38]);

• is non-empty, and

• if and , then there is an element