Speaker
Description
Let $g, f$ be non-negative integer-valued functions on $V(G)$ such that $g(v) \le f(v) \le d_G(v)$ for all $v \in V(G)$.
A $(g,f)$-factor of $G$ is a spanning subgraph $H$ of $G$ such that for every vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$. For $g$ and $f$ with $g(v) \equiv f(v) (\mod 2)$ for all $v \in V(G)$, a $(g, f)$-parity factor of $G$ is a $(g,f)$-factor $H$ such that $d_H(v) \equiv f(v) (\mod 2)$ for all $v \in V(G)$.
For integers $a$ and $b$, an $[a,b]$-factor of $G$ is a $(g,f)$-factor such that $g(v)=a$ and $f(v)=b$ for all $v \in V(G)$,
and a $k$-factor is a $[k,k]$-factor. For odd (or even, respectively) integers $a$ and $b$, an odd (or even, respectively) $[a,b]$-factor is an $[a,b]$-factor $H$ such that $d_H(v)$ is odd (or even, respectively). The eigenvalues of $G$ are the eigenvalues of its adjacency matrix.
In this talk, we investigate eigenvalue conditions for a certain graph to have a $k$-factor, an (even or odd) $[a,b]$-factor,
a $(g,f)$-parity factor, or a connected (even or odd) factor.
