The Alon–Boppana bound is a result in the field of graph theory and spectral graph theory. It provides a lower bound on the largest eigenvalue (also known as the spectral radius) of a regular graph. More formally, let \( G \) be a \( d \)-regular graph on \( n \) vertices.

In graph theory, a **cycle basis** of a graph is a minimal set of cycles such that any cycle in the graph can be expressed as a combination of these cycles. Specifically, for a connected graph, a cycle basis serves as a framework for the cycles of the graph. ### Key Points: 1. **Cycles**: A cycle in a graph is a path that starts and ends at the same vertex, with no other vertices repeated.

A distance-regular graph is a specific type of graph that has a high degree of regularity in the distances between pairs of vertices. Formally, a graph \( G \) is said to be distance-regular if it satisfies the following conditions: 1. **Regularity**: The graph is \( k \)-regular, meaning each vertex has exactly \( k \) neighbors.

Maude is a high-level programming language and system that is based on rewriting logic. It is designed for specifying, programming, and reasoning about systems in a formal and executable manner. Maude allows for the definition of systems in terms of algebraic specifications, and it can be used for a wide range of applications in formal methods, including model checking, theorem proving, and symbolic simulation.

Graph automorphism is a concept in graph theory that refers to a symmetry of a graph that preserves its structure. More specifically, an automorphism of a graph is a bijection (one-to-one and onto mapping) from the set of vertices of the graph to itself that preserves the adjacency relationship between vertices.

The Ihara zeta function is a mathematical object that arises in the study of finite graphs, particularly in the context of algebraic topology and number theory. It was introduced by Yoshio Ihara in the 1960s.

Mac Lane's planarity criterion, also known as the "Mac Lane's formation", is a combinatorial condition used to determine whether a graph can be embedded in the plane without any edges crossing. Specifically, the criterion states that a graph is planar if and only if it does not contain a specific type of subgraph as a "minor.

The Parry–Sullivan invariant is a concept in the field of dynamical systems and statistical mechanics, particularly related to the study of interval exchanges and translations. It is associated with the study of the dynamics of certain classes of transformations, particularly those that exhibit specific structural and statistical properties. The invariant itself is often connected to topological and measure-theoretic characteristics of systems that exhibit a certain type of symmetry or recurrence.

Sims' conjecture is a hypothesis in the field of algebraic topology and combinatorial group theory, specifically relating to the properties of certain types of groups. Named after mathematician Charles Sims, the conjecture primarily deals with the structure of finite groups and representation theory. While specific details or formulations may vary, Sims' conjecture is generally focused on establishing a relationship between the orders of groups and their representations or modules.

A strongly regular graph is a specific type of graph characterized by a regular structure that satisfies certain conditions regarding its vertices and edges. Formally, a strongly regular graph \( G \) is defined by three parameters \( (n, k, \lambda, \mu) \) where: - \( n \) is the total number of vertices in the graph.

A **vertex-transitive graph** is a type of graph in which, for any two vertices, there is some automorphism of the graph that maps one vertex to the other. In simpler terms, this means that the graph looks the same from the perspective of any vertex; all vertices have a similar structural role within the graph. ### Key Properties: 1. **Automorphism:** An automorphism is a bijection (one-to-one correspondence) from the graph to itself that preserves the edges.

As of 2022:

The median household income at the time was 31k^[ref]. Clearly, putting one child through university with that income would be basically impossible, you would pay 19 - 5 = 14k/year, almost half of your income. Two children would be impossible. Remember how each family needs to have two children minimum to perpetuate life?

This book series appears to be the one: global.oup.com/academic/content/series/h/history-of-the-university-of-oxford-huo/. A mere 250 pounds+ each.

For students (who are paying for the university to start with...), they will not claim tutorials linked to courses. But a tutorial that shows university laboratories, it is unclear: www.ox.ac.uk/students/academic/guidance/intellectual-property (archive) This likely includes graduate students, who are also not paid by the university.

For faculty, the university owns everything it seems, to be confirmed.