A New Proof on the Bipartite Turán Number of Bipartite Graphs

The bipartite Turán number of a graph H, denoted by $\text{ex}\left(m,n;H\right)$ , is the maximum number of edges in any bipartite graph $G=\left(A,B;E\left(G\right)\right)$ with $\left|A\right|=m$ and $\left|B\right|=n$ which does not contain H as a subgraph. When$\min \left\{m,n\right\}>2t$ , the problem of determining the value of $\text{ex}\left(m,n;K_{m？t,n？t}\right)$ has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of $\text{ex}\left(m,n;K_{m？t,n？t}\right)$ by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using $\left\{0,1\right\}$ -matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].