Ordering (signless) Laplacian spectral radii with maximum degrees of graphs

Abstract Let λ ( G ) and μ ( G ) be the Laplacian and signless Laplacian spectral radius of a graph G , respectively, and let Δ ( G ) be the maximum degree of G . We call a graph G an ( n , m ) graph if G contains n vertices and m edges. In this paper, we prove that for two connected ( n , m ) graphs G and G ′ , if Δ ( G ) ≥ m − n − 3 2 and Δ ( G ) > Δ ( G ′ ) , then λ ( G ) > λ ( G ′ ) and μ ( G ) > μ ( G ′ ) , and the bound “ m − n − 3 2 ” is optimal for the case of signless Laplacian spectral radius. Moreover, we use an example to illustrate that, as a consequence of our new result, when m ≤ ⌊ 3 n − 5 2 ⌋ , the ordering of connected ( n , m ) graphs according to their largest (signless) Laplacian spectral radii can be transfer to the ordering of connected ( n , m ) graphs with large maximum degree and hence we can conclude that it is not a difficult problem to ordering connected ( n , m ) graphs via their largest (signless) Laplacian spectral radii.