Caroline Grant Melles

Chip-Firing Games

A chip-firing game on a vertex-weighted graph (Gamma ) is a one-player game where a move amounts to selecting a vertex and firing it, i.e., redistributing the weight (or chips) of that vertex among its neighbors. There are a number of variations in this game and we shall discuss some of them below.

Cayley Graphs of Bent Functions and Codes

This chapter introduces some more advanced problems which arise in the intersection of combinatorics, graph theory, error-correcting codes, and cryptography.

Graphs as Manifolds

In this chapter, we are interested in exploring questions such as the following. Question 3.1. Do harmonic maps on a Riemann surface have an analog for graphs? Question 3.2. Does the Riemann–Roch theorem for Riemann surfaces have an analog for graphs? Question 3.3. If there is a harmonic morphism (Gamma _2rightarrow Gamma _1), how are the Laplacians of (Gamma _2, Gamma _1) related? There are several similarities between graphs and manifolds. This chapter explores some of these connections.

Graphs and Laplacians

In this chapter, we are interested in exploring questions such as the following. If a group G acts on a graph (Gamma ), what is the relationship between the spectrum of (Gamma ) and the spectrum of the quotient (Gamma /G)?

Introduction: Graphs—Basic Definitions

By a graph, we don’t mean a chart or function plot or a diagram. We mean a combinatorial structure which consists of vertices, also called points or nodes, connected by edges, also called arcs. If you wish, think of vertices as destinations and edges as routes. A more precise statement is given below.

Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

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