On the Metric Dimension of Arithmetic Graph of a Composite Number

This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of A m is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of A m which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of A m , when m has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of A m , when m has at least three distinct prime divisors.

The Quasi-Isomorphism Theorem

This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.

Graph Master Picture Theorem

This chapter aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.

A Special Case of Completion Invariance for thec2Invariant of a Graph

Thec2invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that thec2invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of thec2invariant in the case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.