Local structure of graphs with \(\lambda= \mu= 2\), \(a_ 2=4\) (Q1906845)

The author studies graphs \(G\) with the following properties: \(\lambda = 2\): For any two adjacent vertices in \(G\), there are exactly two vertices adjacent to both. \(\mu = 2\): For any two vertices at distance 2, there are exactly two vertices adjacent to both. \(a_2 = 4\): For any pair of vertices \(p\), \(q\) of \(G\) at distance 2, there are exactly four vertices adjacent to \(q\) and at distance 2 from \(p\). He proves that the graph induced by the set of neighbors of any vertex in \(G\) is a disjoint union of triangles, hexagons or heptagons. This result implies that there is no distance-regular graph with intersection array (13, 10, 7; 1, 2, 7).