Let G be a graph and let c(x,y) denote the number of vertices in G adjacent to both of the vertices x and y. We call G quadrangular if c(x,y) ~ 1 whenever x and y are distinct vertices in G. Reid and Thomassen proved that IE(G)I >t 21V(G)I -4 for each connected quadrangular graph (7, and characterized those graphs for which the lower bound is attained. Their result implies lower bounds on the number of l's in m x n combinatorially orthogonal (0,1)-matrices, where a (0. I)-matrix A is said to be combinatorially orthogonal if the inner product of each pair of rows and each pair of columns of A is never one. Thus the result of Reid and Thomassen is related to a paper of Beasley, Brualdi and Shader in which they show that a fully indecomposable, combinatorially orthogonal (0,1)-matrix of order n ~ 2 has at least 4n -4 l's and characterize those matrices for which equality holds. One of the results obtained here is equivalent to the result of Beasley, Brualdi and Shader. We also prove that IE(G)I >I 2IV(G)I -I for each connected quadrangular nonbipartite graph G with at least 5 vertices, and characterize the graphs for which the lower bound is attained. In addition, we obtain optimal lower bounds on the number of l's in m x n combinatorially row-orthogonal (0,1)-matrices.