Automated conjectures on upper bounds for the largest Laplacian eigenvalue of graphs

Several upper bounds on the largest Laplacian eigenvalue of a graph G, in terms of degree and average degree of neighbors of its vertices, have been proposed in the literature. We show that all these bounds, as well as many conjectured new ones, can be generated systematically using some simple algebraic manipulations. Bounds depending on the edges of G are also generated. Moreover, the interestingness of bounds is discussed, in terms of dominance and tightness. Finally, we give a unified way of proving a sample of these bounds.