#!/usr/bin/env python # coding: utf-8 # # Nonexistence of a distance-regular graph with intersection array $\{135, 128, 16; 1, 16, 120\}$ # # We will show that a distance-regular graph with intersection array $\{135, 128, 16; 1, 16, 120\}$ does not exist. # In[1]: import drg # Such a graph would have $1360$ vertices. # In[2]: p = drg.DRGParameters([135, 128, 16], [1, 16, 120]) p.order() # We see that all intersection numbers are nonnegative integers. # In[3]: p.show_distancePartitions(vertex_size = 650) # The Krein parameters are also nonnegative. The graph would not be $Q$-polynomial, but we still have $q^3_{33} = 0$. # In[4]: p.kreinParameters() # We check the remaining known feasibility conditions. We skip the sporadic nonexistence check since the intersection array is already included. # In[5]: p.check_feasible(skip = ["sporadic"]) # We now compute the triple intersection numbers with respect to three mutually adjacent vertices $u, v, w$. Note that we have $a_1 = 6$, so such triples must exist. The parameter $a$ will denote the number of vertices adjacent to all of $u, v, w$. # In[6]: S111 = p.tripleEquations(1, 1, 1, params = {"a": (1, 1, 1)}) S111 # The number of vertices at distance $3$ from all of $u, v, w$ should be a nonnegative integer. Nonnegativity is only achieved when $a = 0, 1, 2$, however integrality is not achieved in any of these cases. # In[7]: a = S111[1, 1, 1] [S111[3, 3, 3].subs(a == x) for x in [0, 1, 2, 3]] # We thus conclude that a graph with intersection array $\{135, 128, 16; 1, 16, 120\}$ **does not exist**.