SELECTED TOPICS IN SPECTRAL GRAPH THEORY

Selected Topics in the Extremal Graph Theory

Extremal problems in graph theory form a very wide research area. We study the following topics: the metric dimension of circulant graphs, the Wiener index of trees of given diameter, and the degree‐diameter problem for Cayley graphs. All three topics are connected to the study of distances in graphs. We give a short survey on the topics and present several new results.

Topics in Spectral Theory

2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Lebesgue-Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . ...

Topics in Graph Theory

Abstract. This paper is an exposition of some classic results in graph theory and their applications. A proof of Tutte’s theorem is given, which is then used to derive Hall’s marriage theorem for bipartite graphs. Some compelling applications of Hall’s theorem are provided as well. In the final section we present a detailed proof of Menger’s theorem and demonstrate its power by deriving König’s...

Selected Topics in Intersection Theory

Special Topics in Graph Theory

This three part lecture series is based primarily on the book Elementary Number Theory, Group Theory, and Ramanujan Graphs by Davidoff, Sarnak, and Valette. I will discuss background information and the explicit construction of (p+ 1)-regular Ramanujan graphs by Lubotzky-Phillips-Sarnak and Margulis, where p is an odd prime. 1. Basic definitions and results We start with some basic definitions ...