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

TL;DR

This paper introduces two new families of edge-girth-regular graphs derived from biaffine planes and Suzuki groups, expanding the known extremal and regular graph constructions for specific parameters.

Contribution

The paper constructs novel families of edge-girth-regular graphs using algebraic structures, providing extremal examples and extending the catalog of such graphs for certain parameters.

Findings

First family: extremal $egr(2q^2,q,6,(q-1)^2(q-2))$ for prime power $q extgreater{}3$.

Second family: $egr(q(q^2+1),q,5,\lambda)$ for odd powers of 2 with $\lambda extgreater{}q-1$.

Explicit constructions for specific parameters, including $q=8$, with $\lambda=q-1$.

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 $q\geq 3$ and, the second one is a family of $egr(q(q^2+1),q,5,\lambda)$ for $\lambda\geq q-1$ and $q\geq 8$ an odd power of $2$. In particular, if $q=8$ we have that $\lambda=q-1$.