#!/usr/bin/env python
# coding: utf-8

# # Nonexistence of a distance-regular graph with intersection array $\{104, 70, 25; 1, 7, 80\}$
# 
# We will show that a distance-regular graph with intersection array $\{104, 70, 25; 1, 7, 80\}$ does not exist.

# In[1]:


get_ipython().run_line_magic('display', 'latex')
import drg


# Such a graph would have $1470$ vertices.

# In[2]:


p = drg.DRGParameters([104, 70, 25], [1, 7, 80])
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 be formally self-dual for the natural ordering of eigenspaces and thus also $Q$-polynomial, so we have $q^3_{11} = q^1_{13} = q^1_{31} = 0$.

# 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"])


# Let $w, x, y, z$ be vertices such that $x$ is adjacent to $y$ and $z$, $y$ is at distance $2$ from $w$ and $z$, and $w$ is at distance $3$ from $x$. Note that we have $p^1_{12} = 70$ and $p^1_{32} = 250$, so such vertices must exist. We first compute the triple intersection numbers with respect to $x, y, z$. The parameter $\alpha$ will denote the number of vertices adjacent to $x, y, z$.

# In[6]:


p.tripleEquations(1, 1, 2, params={"alpha": (1, 1, 1)})


# From $[2\ 1\ 1] \ge 0$ and $[3\ 2\ 3] \ge 0$, it follows that there is a single solution with $\alpha = 6$ and therefore $[3\ 2\ 3] = 0$, implying that $w$ cannot be at distance $3$ from $z$ for any choice of $w, x, y, z$ as above.
# 
# We now compute the triple intersection numbers with respect to $w, x, y$. The parameter $\beta$ will denote the number of vertices at distances $(3, 1, 2)$ from $w, x, y$.

# In[7]:


p.tripleEquations(3, 2, 1, params={"beta": (3, 1, 2)})


# From $[3\ 3\ 2] \ge 0$ and $[1\ 3\ 3] \ge 0$, it follows that $15 \le \beta \le 18$. This would imply the existence of a vertex $z$ as above that is at distance $3$ from $w$ - a contradiction! We thus conclude that a distance-regular graph with intersection array $\{104, 70, 25; 1, 7, 80\}$ **does not exist**.