Ordering Planar Graphs by Their Signless Laplacian Spectral Radii

A graph is planar if it can be embedded in the plane such that its edges intersect only at their common endpoints. In this paper, we determine the graphs attaining the second and third largest signless Laplacian spectral radii among all planar graphs of order n≥398. Furthermore, we apply this characterization to identify the planar graphs that achieve the first three largest values of the sum of the first and second largest signless Laplacian eigenvalues.