On Quipus whose signless Laplacian index does not exceed 4.5

Hoffman and Smith proved that in a graph with maximum degree Delta if all edges are subdivided infinitely many times, then the largest eigenvalue, also called index, of the adjacency matrix converges to Delta/(root Delta - 1). For the (signless) Laplacian of graphs, a similar result holds and the limit value is its square Delta(2)/(Delta - 1). Throughout the years, several scholars have progressed into characterizing the (connected) graphs whose adjacency or (signless) Laplacian index does not exceed the Hoffman-Smith limit value for Delta = 3, still there is not a complete characterization of such graphs. Here, we consider the signless Laplacian variant of this problem, and we characterize a large portion of such graphs. Also, we provide a structural restriction for the graphs not yet included for the complete characterization. Finally, we discuss the consequences on the adjacency variant of this problem.