On June 19, 2019, Suil O (오수일) from SUNY Korea, Incheon presented his work on an upper bound of the third largest eigenvalue of a connected r-regular graph to have an odd (1,b)-factor. The title of his talk was “An odd [1,b]-factor in regular graphs from eigenvalues”.
An odd $[1,b]$-factor of a graph is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $1 \le d_H(v) \le b$, and $d_H(v)$ is odd. For positive integers $r \ge 3$ and $b \le r$, Lu, Wu, and Yang gave an upper bound for the third largest eigenvalue in an $r$-regular graph with even number of vertices to guarantee the existence of an odd [1,b]-factor.
In this talk, we improve their bound.