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

Abstract

An edge-girth-regular graph $egr(v,k,g,\lambda )$, is a $k$-regular graph of order $v$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g$-cycles. An $egr(v,k,g,\lambda )$ is called extremal for the triple $(k,g,\lambda )$ if $v$ is the smallest order of any $egr(v,k,g,\lambda )$. In this paper, we introduce two families of edge-girth-regular graphs. The first one is a family of extremal $egr(2q^2,q,6,(q-1{)}^2(q-2))$ for any prime power $q\ge 3$ and, the second one is a family of $egr(q(q^2+1),q,5,\lambda )$ for $\lambda \ge q-1$ and $q\ge 8$ an odd power of $2$. In particular, if $q=8$ we have that $\lambda =q-1$.

17 pages, 1 figure

References in corpus (1)

• Mixed Cages: monotony, connectivity and upper bounds