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4. 3 Two classes of strongly regular graphs - NUI Galway The graph C5 does belong to an infinite family of strongly regular graphs known as the Paley graphs, which are constructed from finite fields A Paley graph on p vertices exists for every p with the property that p is a power of some prime and p 1 mod 4
4. 2 The Adjacency Spectrum of a strongly regular graph In this section we use the defining properties of a strongly regular graph to show that the adja-cency matrix of such a graph satisfies a particular quadratic equation, from which we deduce that the adjacency spectrum can have at most three distinct elements
The Laplacian Matrix of a Graph - NUI Galway The Laplacian matrix of a graph carries the same information as the adjacency matrix obvi-ously, but has different useful and important properties, many relating to its spectrum We start with a few examples
Strongly Regular Graphs - NUI Galway Recall that a (simple, undirected) graph is regular if all of its vertices have the same degree This is a strong property for a graph to have, and it can be recognized easily from the adjacency matrix, since it means that all row sums are equal, and that 1 is an eigenvector
Subgroups - NUI Galway It consists of all the permutations of {1, 2, 3} (with 4 fixed) It is a “copy” of S3 inside S4 In the dihedral group D2n (the symmetries of a regular n-gon), the set of rotational symmetries is a subgroup The set of relections is not (Why?)
1. 2 Some matrix background - maths. nuigalway. ie This is the only regular graph satisfying the hypothesis of the theorem, and it also satisfies the conclusion (and it is a windmill) By the first half of the proof, every non-regular graph that possesses the friendship propery has a vertex adjacent to all others, so we have proved the theorem
MA3343 Groups - Introduction - Lecture 1 Monday Tuesday - email to all students with updates and plan for the week Thursday Friday - lectures in person on campus (we will insist on face coverings and open windows) Tutorials (in person and or online) will begin in Week 3 or 4 There we will discuss current content; look at homework problems;
Bounded Sequences - NUI Galway Give and or identify examples of sequences with or without various properties (or combinations of properties) from the above list; State, prove and apply the Monotone Convergence Theorem; Analyze examples similar to Example 83
Real Symmetric Matrices - NUI Galway In general, if a graph G is regular of degree k, then L(G) will be regular of degree 2k − 2 For any graph G, a vertex of degree d in G corresponds to a copy of the complete graph Kd within L(G)