Signless Laplacian spectral characterization of some joins

Abstract

The join of two disjoint graphs G and H, denoted by G ⨠H, is the graph obtained by joining each vertex of G to each vertex of H. In this paper, the signless Laplacian characteristic polynomial of the join of two graphs is first formulated. And then, a lower bound for the i-th largest signless Laplacian eigenvalue of a graph is given. Finally, it is proved that G ⨠K_m, where G is an (n â 2)-regular graph on n vertices, and K_n ⨠K_2 except for n = 3, are determined by their signless Laplacian spectra.