On some extremal properties of bipartite graphs of large girt

This dissertation is concerned with some extremal properties of bipartite graphs of large girth. A near-linear space has a representation as a bipartite point-line incidence graph so it is natural to consider the extremal properties of such a graph. Every bipartite graph with bipartition classes of sizes m and n, $nge m,$ may be embedded in a bipartite graph with bipartition classes of equal cardinality n. Therefore, with the exception of one theorem in Chapter 2, we concentrate on the properties of bipartite graphs, where the bipartition has parts of equal size.;Define $G=G(n,n)$ to be a simple, bipartite graph of girth g with partition classes of equal cardinality n and e edges. It has been shown in (Bol79) that for $gge 6$ the number of edges of G is bounded above by $rcdot n$ where r is the positive root of the equation $xsp2 - x+1 - n=0.$ C. P. Teo and K. M. Koh proved in (TK92) that if G is 2-connected the number of cycles of length 6 is bounded above by ${eover6}(e - 2n+1).$ Equality holds in both instances if and only if G is the point-line incidence graph of a finite projective plane of order $q=r - 1.$ We will provide a proof of a bound on the number of 6-cycles that depends only on n using these two previous results that does not require that G be 2-connected. A second proof of this bound is presented as a corollary to a theorem on a bound on the number of 6-cycles for the larger class of bipartite graphs with partition classes of cardinality n, m and girth at least six. This second approach does not rely on the previous results.;Furthermore, for $gge 8$ we show under certain conditions that the number of edges of G is bounded above by $rcdot n$ where r is the positive real root of the equation $xsp3 - 2xsp2+2x - n=0.$ As a consequence of this we also obtain an upper bound on the number of cycles of length eight in G with equality in both cases if and only if G is a generalized 4-gon (see definition immediately preceding Theorem 1.2). In addition, we draw some conclusions on a bound for the number of edges if these conditions do not hold. We also speculate on the existence of similar bounds for bipartite graphs of girth $gge 10.$.