Edge-girth-regular graphs arising from biaffine planes and Suzuki groups

Abstract

An edge-girth-regular graph egr(v,k,g,λ), is a k-regular graph of order v, girth g and with the property that each of its edges is contained in exactly λ distinct g-cycles. An egr(v,k,g,λ) is called extremal for the triple (k,g,λ) if v is the smallest order of any egr(v,k,g,λ). In this paper, we introduce two families of edge-girth-regular graphs. The first one is a family of extremal egr(2q2,q,6,(q−1)2(q−2)) for any prime power q≥3. The second one is a family of egr(q(q2+1),q,5,λ) for λ≥q−1 and q≥8 an odd power of 2. In particular, if q=8 we have that λ=q−1. Finally, we construct two egr(32,5,5,12) and we prove that they are extremal.