On biregular bipartite graphs of small excess

Let m > n ≥ 2 and g ≥ 4 be positive integers. A graph G is called a bipartite ( m , n ; g ) -graph if it is a biregular bipartite graph of even girth g with the additional property that all vertices in each of the two partition sets are of the same degree; m in one of them, and n in the other. In analogy with the well-known Cage Problem, if we let v ( G ) denote the order of G , and let B ( m , n ; g ) denote the natural lower bound for the order of bipartite ( m , n ; g ) -graphs obtained as a generalization of the Moore bound for regular graphs, we call the difference v ( G ) − B ( m , n ; g ) the excess of G . The focus of this paper is on the study of the question of the existence of bipartite ( m , n ; g ) -graphs for given parameters ( m , n ; g ) and excess at most 4. We prove that such graphs are rare by finding restrictive necessary arithmetic conditions on the parameters m , n and g . Furthermore, we prove the non-existence of bipartite ( m , n ; g ) -graphs of excess at most 4 for all parameters m , n , g where g ≥ 10 and is not divisible by 4, and m > n ≥ 3 . In the case when the girth of G is 6, we employ spectral analysis of the distance matrices of G , and find necessary relations between their eigenvalues. Finally, we prove for all pairs m , n , m > n ≥ 3 , that the asymptotic density of the set of even girths g ≥ 8 for which there exists a bipartite ( m , n ; g ) -graph with excess not exceeding 4 is equal to 0.