Eigenvalue bounds of the Kirchhoff Laplacian

We prove the inequality ${\lambda }_k\le d_k+d_{k-1}$ for all the eigenvalues ${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n$ of the Kirchhoff matrix K of a finite simple graph or quiver with vertex degrees $d_1\le d_2\le \cdots \le d_n$ and assuming $d_0=0$. Without multiple connections, the inequality ${\lambda }_k\ge \max (0,d_k-(n-k\left)\right)$ holds. A consequence in the finite simple graph or multi-graph case is that the pseudo determinant $\mathrm{Det}\left(K\right)$ counting the number of rooted spanning trees has an upper bound $2^n{\prod }_{k=1}^n{d}_k$ and that $\det (1+K)$ counting the number of rooted spanning forests has an upper bound ${\prod }_{k=1}^n(1+2d_k)$.