An Incomparable Upper Bound for the Largest Laplacian Graph Eigenvalue

Abstract

In this paper, we obtain the following upper bounds for the largest Laplacian graph eigenvalue: $$\mu \le \underset{i}{max}\left\{\sqrt{2d_i(m_i+d_i)+n–2d_i–2\sum _{j:j\sim i}|N_i\cap N_j|}\right\}$$ where $d_i$ and $m_i$ are the degree of vertex $i$ and the average degree of vertex $i$, respectively; $|N_i\cap N_j|$ is the number of common neighbors of vertices $i$ and $j$. We also compare this bound with some known upper bounds.