Counting odd length cycles in regular graphs

Abstract

This thesis explores the enumeration and structural analysis of cycles in graphs, with a particular focus on odd-length cycles in both regular and general graphs. The study is motivated by the combinatorial complexity inherent in cycle structures and their significance in various applications across graph theory and network analysis. A review of existing literature is presented, covering methods for counting cycles in bipartite graphs and general graphs, as well as highlighting the challenges posed by isomorphism checks in large graph structures. Building on this foundation, the thesis develops and implements an algorithm to generate and extend cycle graphs systematically. This algorithm is applied to count and classify odd-length cycles in regular graphs. Although e!ective for smaller girth values, the algorithm’s scalability is limited due to the exponential growth in the number of graph structures and the computational cost of isomorphism detection. Further, the thesis investigates the enumeration of odd-length cycles in general graphs and identifies key limitations in using purely combinatorial approaches to derive closed-form expressions for the number of such structures. These challenges point to the need for alternative strategies, including more e”cient algorithms or algebraic methods, to extend the current framework. Overall, the thesis contributes both algorithmic techniques and theoretical insights to the field of cycle enumeration in graph theory, and it lays the groundwork for future research into more scalable and generalizable methods.