Is this definition of a complete bipartite graph correct? The book I am using defined a complete graph $K_{m,n} = \overline{K}_m + \overline{K}_n$. Is this correct? I am confused since the complement of a complete graph is an empty graph. How is the addition of two empty graph a complete bipartite graph if we follow the definition?

It seems like in your book the notation $G_1 + G_2$ refers to the graph obtained as follows: take the graphs $G_1$ and $G_2$ and join each vertex in $G_1$ to each vertex in $G_2$.  
Thus, if we take the empty graph on $m$ vertices $x_1,\ldots,x_m$ and the empty graph on $n$ vertices $y_1,\ldots,y_n$ and join each $x_i$ to each $y_j$, we get the graph $K_{m,n}$. 
