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An odd $$-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 …

For positive integers, $r\ge 3,h\ge 1,$ and $k\ge 1$, Bollobás, Saito, and Wormald proved some sufficient conditions for an $h$-edge-connected $r$-regular graph to have a k-factor in 1985. Lu gave an upper bound for the third-largest eigenvalue in a connected $r$-regular graph to have a $k$-factor in 2010. Gu found an upper bound …