Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3

Abstract

The study of graphs with the property specified in the title has earlier been reduced to the case of neighborhoods with parameters (35, 18, 9, 9), (36, 21, 12, 12), (40, 27, 18,18), (50, 28, 15, 16), (56, 45, 36, 36), and (64, 27, 10, 12). It is proved that a completely regular graph in which the neighborhoods of vertices are strongly regular graphs with the corresponding parameters is either a strongly regular graph with parameters (76, 35, 18, 14) or (64, 35, 18, 20) or a Taylor graph with intersection array {35, 16, 1; 1, 16, 35} (this is the main theorem). As a consequence, a classification of distance regular graphs in which the neighborhoods of vertices are strongly regular graphs with a non principal eigenvalue r, 2 < r ≤ 3, is obtained.