On girth-biregular graphs

Let Γ denote a finite, connected, simple graph. For an edge e of let n(e) the number girth cycles containing e. vertex v {e1, e2, …, ek} be set edges incident to ordered such that n(e1) ≤ n(e2) … n(ek). Then (n(e1),n(e2),…,n(ek)) is called signature v. The graph said girth-biregular if it bipartite, and all its vertices belonging same bipartition have signature. with g = 2d signatures (a1,a2,…,ak1) (b1,b2,…,bk2), assume without loss generality k1 ≥ k2. Our first result {a1, a2, ak1} {b1, b2, bk2}. next upper bound ak1 M, where M (k1−1)⌊g/4⌋(k2−1)⌈g/4⌉. We describe graphs attaining equality. d 3 or 4 even they are incidence Steiner systems generalized polygons, respectively. Finally, we show when − ε for some non-negative integer < k2 1, then 0. Similar valid 3, 1 k2|̸ k1.