#!/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**.