THE SANDPILE & SMITH GROUPS OF CERTAIN CLASSES OF GRAPHS

The Smith group and the sandpile group are graph invariants. In this thesis we compute these groups for a variety of classes of graphs. We provide background materials in Chapter 1. In Chapter 2, on integral circulant graphs, Theorem A provides the Smith group of multiple copies of these graphs. For integral circulant graphs of prime power order, Theorem B presents a new graph construction that is isomorphic to them. For a subset of these graphs, the Smith group of the Kronecker product of a graph with an all ones matrix is presented in Theorem C. The sandpile group of another subset is given in Theorem D; Conjecture 2.3.6 proposes a generalisation. Theorem E provides the maximum number of integral circulant graphs of order n with d components where d divides n. In Chapter 3, we focus on graphs with at most four distinct eigenvalues, including a number of symmetric graphs. Theorems F, G gives the sandpile group structure of a Kronecker product of the complement of a 6-cycle with an all ones matrix and its complement. Theorems H,I gives the sandpile group of modified complete bipartite graphs. Chapter 4 is about the sandpile group of some classes of threshold graph, and other graph constructions that were selected using Maple code. They are covered in Theorems J, K, L. In Chapter 5, we consider graphs with sandpile group of small rank, giving a condition for a graph join between a path and a complete graph of order two to have non-cyclic sandpile group in Theorem M, and discussing when it is cyclic in Conjecture 5.2.18. Finally, as many theorems that we prove were conjectured by running experiments in Maple, in Chapter 6 we describe the computational approach we used and provide code and examples to facilitate future research.