  An edge-girth-regular graph $egr(n,k,g,\lambda)$ is a $k-$regular graph of
order $n$, girth $g$ and with the property that each of its edges is contained
in exactly $\lambda$ distinct $g-$cycles. We present new families of edge-girth
regular graphs arising from generalized quadrangles and pencils of elliptic
quadrics. An $egr(n, k, g, \lambda)$ is called extremal for the triple $(k, g,
\lambda)$ if $n$ is the smallest order of any $egr(n, k, g, \lambda)$. We give
new lower bounds for the order of extremal edge-girth-regular graphs using
properties of the eigenvalues of the adjacency matrix of a graph.
