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: \[\mu \leq \max\limits_{i} \left\{\sqrt{ 2d_i (m_i + d_i) + n – 2d_i – 2 \sum\limits_{j:j\sim i}{ |N_i \cap N_j|}} \right\}\] 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.