#!/usr/bin/env python # coding: utf-8 # # Nonexistence of a distance-regular graph with intersection array $\{234, 165, 12; 1, 30, 198\}$ # # We will show that a distance-regular graph with intersection array $\{234, 165, 12; 1, 30, 198\}$ does not exist. # In[1]: import drg # Such a graph would have $1600$ vertices. # In[2]: p = drg.DRGParameters([234, 165, 12], [1, 30, 198]) 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. We have $q^1_{22} = q^2_{12} = q^2_{21} = 0$, so the graph would be $Q$-polynomial with respect to the ordering of the eigenvalues $0, 2, 3, 1$. # 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 vertices $u, v, w$ at mutual distances $3$. Note that we have $p^3_{33} = 8$, so such triples must exist. The parameter $a$ will denote the number of vertices at distance $3$ from all of $u, v, w$. # In[6]: S333 = p.tripleEquations(3, 3, 3, params = {"a": (3, 3, 3)}) S333 # We now note that since $a$ must be nonnegative, the number of vertices at distances $2, 3, 3$ from $u, v, w$ must be negative - contradiction # In[7]: print(S333[2, 3, 3]) print(S333[3, 3, 3]) # We thus conclude that a graph with intersection array $\{234, 165, 12; 1, 30, 198\}$ **does not exist**.