6 problems

Let $egr(v,k,g,\lambda)$ denote an edge-girth-regular graph with $v$ vertices, valency $k$, girth $g$, and each edge contained in exactly $\lambda$ cycles of length $g$. Let…

Let $n(k,g,\lambda)$ denote the minimum order of an edge-girth-regular graph with valency $k$, girth $g$, and edge-girth-regularity parameter $\lambda$, with…

Let $q\ge 3$ be a prime power and let $g\in\{8,12\}$. An $egr(v,k,g,\lambda)$-graph is a $k$-regular graph of order $v$ and girth $g$ in which every edge is contained in $\lambda$…

Higher-girth existence conjecture. For $q\geq 3$ a prime power and $g\in\{8,12\}$, there exists a family of

Smallness conjecture. The construction given in Section $$ produces small girth-regular graphs, meaning graphs whose order is close to being extremal.

The edge-girth conjecture. The family of graphs $\mathcal G_q(V,E)$ is a family of