On Crown-Free Graphs with Regular $\mu$-Subgraphs, II

The crown is the complete multipartite graph $K_{1,1,3}$. Terwilliger graphs without crowns and graphs without 3-cocliques with regular $\mu$-subgraphs of given positive degree are studied. As a corollary, the local structure of graphs in which the neighborhoods of vertices are regular Terwilliger graphs of diameter 2 and some of these neighborhoods contain no 7-paws is determined. Connected crown-free graphs in which $\mu$-subgraphs are edge regular graphs of diameter not exceeding 2 with given parameters are described.