We will show that a distance-regular graph with intersection array $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$ does not exist. The existence of such a graph would give a counterexample to a conjecture by MacLean and Terwilliger, see Bipartite distance-regular graphs: The $Q$-polynomial property and pseudo primitive idempotents by M. Lang.
import drg
Such a graph would be bipartite with $3500$ vertices.
p = drg.DRGParameters([55, 54, 50, 35, 10], [1, 5, 20, 45, 55])
print(p.is_bipartite())
print(p.order())
True 3500
We see that all intersection numbers are nonnegative integers.
p.show_distancePartitions(vertex_size = 650)
The Krein parameters are also nonnegative. Many Krein parameters are zero - in particular, the graph would be $Q$-polynomial for the natural ordering of the eigenvalues.
p.kreinParameters()
0: [ 1 0 0 0 0 0] [ 0 132 0 0 0 0] [ 0 0 1617 0 0 0] [ 0 0 0 1617 0 0] [ 0 0 0 0 132 0] [ 0 0 0 0 0 1] 1: [ 0 1 0 0 0 0] [ 1 50/3 343/3 0 0 0] [ 0 343/3 2450/3 686 0 0] [ 0 0 686 2450/3 343/3 0] [ 0 0 0 343/3 50/3 1] [ 0 0 0 0 1 0] 2: [ 0 0 1 0 0 0] [ 0 28/3 200/3 56 0 0] [ 1 200/3 2380/3 700 56 0] [ 0 56 700 2380/3 200/3 1] [ 0 0 56 200/3 28/3 0] [ 0 0 0 1 0 0] 3: [ 0 0 0 1 0 0] [ 0 0 56 200/3 28/3 0] [ 0 56 700 2380/3 200/3 1] [ 1 200/3 2380/3 700 56 0] [ 0 28/3 200/3 56 0 0] [ 0 0 1 0 0 0] 4: [ 0 0 0 0 1 0] [ 0 0 0 343/3 50/3 1] [ 0 0 686 2450/3 343/3 0] [ 0 343/3 2450/3 686 0 0] [ 1 50/3 343/3 0 0 0] [ 0 1 0 0 0 0] 5: [ 0 0 0 0 0 1] [ 0 0 0 0 132 0] [ 0 0 0 1617 0 0] [ 0 0 1617 0 0 0] [ 0 132 0 0 0 0] [ 1 0 0 0 0 0]
We check the remaining known feasibility conditions. We skip the sporadic nonexistence check since the intersection array is already included.
p.check_feasible(skip = ["sporadic"])
We now compute the triple intersection numbers with respect to three vertices $u, v, w$ at mutual distances $2$. Note that we have $p^2_{22} = 5$, so such triples must exist. The parameter $a$ will denote the number of vertices adjacent to all of $u, v, w$.
S222 = p.tripleEquations(2, 2, 2, params = {"a": (1, 1, 1)})
S222
0: [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 1 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] [0 0 0 0 0 0] 1: [ 0 0 0 0 0 0] [ 0 a 0 -a + 5 0 0] [ 0 0 0 0 0 0] [ 0 -a + 5 0 a + 45 0 0] [ 0 0 0 0 0 0] [ 0 0 0 0 0 0] 2: [ 0 0 1 0 0 0] [ 0 0 0 0 0 0] [ 1 0 20*a + 92 0 -20*a + 150 0] [ 0 0 0 0 0 0] [ 0 0 -20*a + 150 0 20*a + 200 0] [ 0 0 0 0 0 0] 3: [ 0 0 0 0 0 0] [ 0 -a + 5 0 a + 45 0 0] [ 0 0 0 0 0 0] [ 0 a + 45 0 11*a + 1055 0 -12*a + 160] [ 0 0 0 0 0 0] [ 0 0 0 -12*a + 160 0 12*a + 15] 4: [ 0 0 0 0 0 0] [ 0 0 0 0 0 0] [ 0 0 -20*a + 150 0 20*a + 200 0] [ 0 0 0 0 0 0] [ 0 0 20*a + 200 0 -20*a + 605 0] [ 0 0 0 0 0 0] 5: [ 0 0 0 0 0 0] [ 0 0 0 0 0 0] [ 0 0 0 0 0 0] [ 0 0 0 -12*a + 160 0 12*a + 15] [ 0 0 0 0 0 0] [ 0 0 0 12*a + 15 0 -12*a + 20]
The number of vertices at distance $5$ from all of $u, v, w$ is a nonnegative integer, implying that the parameter $a$ is either $0$ or $1$.
print(S222[1, 1, 1])
print(S222[5, 5, 5])
a -12*a + 20
Let us now consider the set $A$ of common neighbours of $u$ and $v$, and the set $B$ of vertices at distance $2$ from both $u$ and $v$. Their sizes are given by the intersection numbers $p^2_{11}$ and $p^2_{22}$.
print(p.p[2, 1, 1])
print(p.p[2, 2, 2])
5 243
By the above, each vertex in $B$ has at most one neighbour in $A$, so there are at most $243$ edges between $A$ and $B$. However, each vertex in $A$ is adjacent to both $u$ and $v$, and the other $k - 2 = 53$ neighbours are in $B$, amounting to a total of $5 \cdot 53 = 265$ edges. We have arrived to a contradiction, and we must conclude that a graph with intersection array $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$ does not exist.