id: 05827408
dt: j
an: 05827408
au: Lu, Hongliang
ti: Regular factors of regular graphs from eigenvalues.
so: Electron. J. Comb. 17, No. 1, Research Paper R159, 12 p., electronic only
    (2010).
py: 2010
pu: Prof. André Kündgen, Deptartment of Mathematics, California State
    University San Marcos, San Marcos, CA
la: EN
cc: 
ut: lower bound; best upper bound; third largest eigenvalue; second largest
    eigenvalue
ci: Zbl 1205.05177
li: emis:journals/EJC/Volume_17/Abstracts/v17i1r159.html
ab: Summary: Let $r$ and $m$ be two integers such that $r\ge m$. Let $H$ be a
    graph with order $|H|$, size $e$ and maximum degree $r$ such that
    $2e\ge |H|r-m$. We find a best lower bound on the spectral radius of a
    graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular
    graph of order $|G|$ and $k<r$ be an integer. Using the previous
    results, we find some best upper bounds (in terms of $r$ and $k$) on
    the third largest eigenvalue that is sufficient to guarantee that $G$
    has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on
    the second largest eigenvalue that is sufficient to guarantee that $G$
    is $k$-critical when $k|G|$ is odd. Our results extend the work of {\it
    S.M. Cioabă}, {\it D.A. Gregory} and {\it W.H. Haemers} [J. Comb.
    Theory, Ser. B 99, No.\,2, 287‒297 (1999; Zbl 1205.05177)] who
    obtained such results for 1-factors.
rv: