Maximizing the signless Laplacian spectral radius of minimally 3-connected graphs with given size

Abstract

A 3-connected graph G is minimally 3-connected if it becomes not 3-connected after deleting any edge of G. In this paper, we give an upper bound on the signless Laplacian spectral radius of minimally 3-connected graphs with m edges, and completely characterize the corresponding extremal graph in the case when m is even. As a corollary, we determine the unique graph with the maximal signless Laplacian spectral radius among all Halin graphs with m edges.