A new estimate for the vertex number of an edge-regular graph

Given a connected edge-regular graph $\Gamma$ with parameters $(v,k,\lambda)$ and $b_1=k-\lambda-1$, we prove that in the case $k\geqslant3b_1-2$ either $|\Gamma_2(u)|(k-2b_1+2)<kb_1$ for every vertex $u$ or $\Gamma$ is a polygon, the edge graph of a trivalent graph without triangles that has diameter greater than 2, the icosahedral graph, the complete multipartite graph $K_{r\times2}$, the $3\times3$-grid, the triangular graph $T(m)$ with $m\leqslant7$, the Clebsch graph, or the Schläfli graph.