Extremal Edge-Girth-Regular Graphs

Abstract

An edge-girth-regular $egr(v,k,g,\lambda )$-graph $\Gamma$ is a k-regular graph of order v and girth g in which every edge is contained in $\lambda$ distinct g-cycles. Edge-girth-regularity is shared by several interesting classes of graphs which include edge- and arc-transitive graphs, Moore graphs, as well as many of the extremal k-regular graphs of prescribed girth or diameter. Infinitely many $egr(v,k,g,\lambda )$-graphs are known to exist for sufficiently large parameters $(k,g,\lambda )$, and in line with the well-known Cage Problem we attempt to determine the smallest graphs among all edge-girth-regular graphs for given parameters $(k,g,\lambda )$. To facilitate the search for $egr(v,k,g,\lambda )$-graphs of the smallest possible orders, we derive lower bounds in terms of the parameters k, g and $\lambda$. We also determine the orders of the smallest $egr(v,k,g,\lambda )$-graphs for some specific parameters $(k,g,\lambda )$, and address the problem of the smallest possible orders of bipartite edge-girth-regular graphs.