This paper consists of two parts. Firstly, we study the upper bound on the signless Laplacian spectral radius of simple graphs and planar graphs. For a graph, if there exists an orientation such that the maximum out-degree Δ+ is no more than half of the maximum degree Δ, by estimating the number of the semi-edge walks of the graph, we obtain a new upper bound of the signless Laplacian spectral radius of the graph, that is Moreover, combining with the result of the edge decomposition of a planar graph by D.Goncalves, a new upper bound of the signless Laplacian spectral radius of a planar graph is presented, that is,Secondly, we study the upper bound on the minimum balanced bipartition of simple graphs and planar graphs. Given a graph, we use the adjacency matrix of the graph to express the number of edge cuts of a balanced bipartition, and obtain a new upper bound on the minimum balanced bipartition, that is, m and n are the number of edges and vertices of the graph, respectively. Moreover, for a planar graph, if the number of edges is less than twice of the number of vertices, then there must exist a balanced bipartition such that the number of edge cuts is no more than the number of vertices. |