Strongly regular graphs with \(\lambda\) = 1 (Q1813197)

For a graph \(\Gamma\) with \(\lambda=1\) we define a graph \(\Gamma_ \Delta\) whose vertices are the \(\Delta\)-delta triangles of \(\Gamma\), and two triangles are connected by an edge in \(\Gamma_ \Delta\) if they have exactly one common vertex. Then \(\Gamma_ \Delta\) is an edge regular graph with parameters \[ v_ \Delta=vk/6,\quad k_ \Delta=3(k-2)/2,\quad \lambda_ \Delta=(k-2)/2-1. \] Von der Flaass showed that if \(\Gamma_ \Delta\) is strongly regular, then \(k=2\mu\). He showed that if \(\Gamma\) is a strongly regular graph with \(\lambda=1\) and \(k=2\mu\), then \(\mu=2,3\) or 5 and \(\Gamma\) is a uniquely determined graph with parameters (9, 4, 1, 2), (15, 6, 1, 3) or (27, 10, 1, 5). These graphs are graphs of rank 3 and their automorphism groups are isomorphic to \(E_ 9\cdot D_ 8\), \(\Sigma_ 6\) and \(O^ -_ 6(2)\) respectively. Besides these graphs, the existence of three other graphs with \(\lambda=1\) is known. They have parameters (81, 20, 1, 6), (243, 22, 1, 2) and (729, 112, 1, 20). We weaken the condition of strong regularity of the graph \(\Gamma_ \Delta\) to the following condition: (*) any pair of triangles of \(\Gamma\), joined by at least two edges, is connected by exactly three edges. Theorem 1. Let \(\Gamma\) be a strongly regular graph with \(\lambda=1\) satisfying the condition (*). Then either \(\mu\leq 3\), or \(\Gamma\) is a graph with parameters (27, 10, 1, 5). The least parameters of graphs with \(\lambda=1\), \(\mu\leq 3\) and \(k\neq2\mu\) are (99, 14, 1, 2) and (115, 18, 1, 3). Theorem 2. There are no strongly regular graphs with parameters (99, 14, 1, 2) and (115, 18, 1, 3) satisfying condition (*). Theorem 2 gives a partial answer to Seidel's question on the existence of a strongly regular graph with parameters (99, 14, 1, 2).