We study extremal and structural problems in regular graphs involving various parameters. In Chapter 2, we obtain the best lower bound for the matching number over $n$-vertex connected regular graphs in terms of edge-connectedness and determine when the matching number is minimized.We also establish the best upper bound for the number of cut-edges over $n$-vertex connected odd regular graphs and determine when the number of cut-edges is maximized. In addition, there is a relationship between the matching number and the total domination number in regular graphs.In Chapter 3, we explore the relationship between eigenvalue and matching number in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a $l$-edge-connected $d$-regular graph, when $l\leq d-2$. We also study what is the weakest hypothesis on the second largest eigenvalue $\lambda_2$ for a $d$-regular graph $G$ to guarantee that $G$ is $l$-edge-connected.In Chapter 4, we study several extremal problems for regular graphs, including the Chinese postman problem, the path cover number, the average edge-connectivity, and the number of perfect matchings. In Chapter 5, we study an $r$-dynamic coloring problem and give the relationship between the $r$-dynamic chromatic number and the chromatic number in regular graphs. We also study $r$-dynichromatic number of the cartesian product of paths and cycles.
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Matchings, Connectivity, and Eigenvalues in Regular Graphs
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