Let G be a simple graph with order n and adjacency matrix A (G) . The characteristic polynomial of G is defined by ϕ (G ; λ) = det (λ I - A (G)) = ∑ i = 0 n a i (G) λ n - i , where a i (G) is called the i-th adjacency coefficient of G. Denote by B n , m the collection of all connected bipartite graphs having n vertices and m edges. A bipartite graph G is referred as 4-Sachs optimal if a 4 (G) = min { a 4 (H) ∣ H ∈ B n , m }. For any given integer pair (n, m), in this paper we investigate the 4-Sachs optimal bipartite graphs. Firstly, we show that each 4-Sachs optimal bipartite graph is a difference graph. Then we deduce some structural properties on 4-Sachs optimal bipartite graphs. Especially, we determine the unique 4-Sachs optimal bipartite (n, m)-graphs for n ≥ 5 and n - 1 ≤ m ≤ 2 (n - 2) . Finally, we provide a method to construct a class of cospectral difference graphs, which disprove a conjecture posed by Andelić et al. (J Czech Math 70:1125–1138, 2020). [ABSTRACT FROM AUTHOR]