EDGE BIPARTITENESS AND SIGNLESS LAPLACIAN SPREAD OF GRAPHS

Let G be a connected graph, and let epsilon(b)(G) and S-Q(C) be the edge bipartiteness and the signless Laplacian spread of G, respectively. We establish some important relationships between epsilon(b)(G) and S-Q(G), and prove S-Q (G) >= 2(1 + cos pi/n), with equality if and only if G = P-n or G = C-n in case of odd n. In addition, we show that if G not equal P-n or G not equal C2k+1, then S-Q(G) >= 4, with equality if and only if G is one of the following graphs: K-1,K-3, K-4, two triangles connected by an edge, and C-n for even n. As a consequence, we prove a conjecture of CVETKOVIC, ROWLINSON and SIMIC on minimal signless Laplacian spread [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), 81 (95) (2007), 11-27].