Let $G$ be a graph of order $n$ with signless Laplacian eigenvalues $q_1, ldots,q_n$ and Laplacian eigenvalues $mu_1,ldots,mu_n$. It is proved that for any real number $lpha$ with $0 lphaleq1$ or $2leqlpha 3$, the inequality $q_1^lpha+cdots+ q_n^lphageq mu_1^lpha+cdots+mu_n^lpha$ holds, and for any real number $eta$ with $1 eta 2$, the inequality $q_1^eta+cdots+ q_n^etale mu_1^eta+cdots+mu_n^eta$ holds. In both inequalities, the equality is attained (for $lpha otin {1,2}$) if and only if $G$ is bipartite.
展开▼
