On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

Abstract

— We consider Q -polynomial graphs of diameter 4 . Alongside the in fi nite series of intersection arrays { m (2 m + 1) , ( m − 1)(2 m + 1) , m 2 , m ; 1 , m, m − 1 , m (2 m + 1) } , the following admissible intersection arrays of Q -polynomial graphs of diameter 4 with at most 4096 vertices are known: { 5 , 4 , 4 , 3; 1 , 1 , 2 , 2 } (the odd graph on 9 vertices), { 9 , 8 , 7 , 6; 1 , 2 , 3 , 4 } (the folded 9 -cube), { 36 , 21 , 10 , 3; 1 , 6 , 15 , 28 } (the half 9 -cube), and { 53 , 40 , 28 , 16; 1 , 4 , 10 , 28 } . We prove that there is no distance-regular graphs with intersection array { 53 , 40 , 28 , 16; 1 , 4 , 10 , 28 } . DOI