Edge Bipartiteness and Signless Laplacian Spread of Graphs

Let $G$ be a connected graph, and let $eb(G)$ and $sq(G)$ be the edge bipartiteness and the signless Laplacian spread of $G,$ respectively. We establish some important relationships between $eb(G)$ and $sq(G),$ and prove $ sq(G) ge 2 Big(1+cos fracc{pi}{n} Big),$ with equality if and only if $G=P_n$ or $G=C_n$ in case of odd $n.$ In addition, we show that if $G eq P_n$ or $G eq C_{2k+1},$ then $sq(G) ge 4,$ with equality if and only if $G$ is one of the following graphs: $K_{1,3},$ $K_4,$ two triangles connected by an edge, and $C_n$ for even $n.$ As a consequence, we prove a conjecture of extsc{Cvetkovi'c, Rowlinson} and extsc{Simi'c} on minimal signless Laplacian spread [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), {f81} (95) (2007), 11--27].