Adjacency-matrix form of Conway's 99-graph problem: $A^2+A=12I+2J$

Conway's 99-graph problem asks for a strongly regular graph with parameters $(99,14,1,2)$. This statement is the adjacency-matrix form of that existence question.

The assertion is that there exists a $99\times 99$ matrix $A=(a_{ij})$ with natural number entries such that

1. $a_{ij}\in\{0,1\}$ for all $i,j$;
2. $a_{ij}=a_{ji}$ for all $i,j$;
3. $a_{ii}=0$ for all $i$;
4. for all $i,j$,

that is, $A^{2}+A=12I+2J$ with $I$ the identity matrix and $J$ the all-ones matrix.

A symmetric $0/1$ matrix with zero diagonal is exactly the adjacency matrix of a simple graph $G$ on $99$ vertices, and $(A^{2})_{ij}$ counts the common neighbours of $i$ and $j$. On the diagonal, condition 4 reads $(A^{2})_{ii}=14$, which is the degree of vertex $i$, so $G$ is $14$-regular. Off the diagonal it splits into two cases: if $i$ and $j$ are adjacent then $a_{ij}=1$ and $(A^{2})_{ij}=1$, so adjacent vertices have exactly one common neighbour; if they are distinct and non-adjacent then $a_{ij}=0$ and $(A^{2})_{ij}=2$, so non-adjacent vertices have exactly two common neighbours. The four conditions therefore hold for some $A$ if and only if a strongly regular graph with parameters $(99,14,1,2)$ exists.

The formulation is the specialisation to $(n,k,\lambda,\mu)=(99,14,1,2)$ of the standard identity $A^{2}=kI+\lambda A+\mu (J-I-A)$ characterising strongly regular graphs. It states the problem as a finite arithmetic condition on a $0/1$ matrix, the form in which computational searches and orbit-matrix arguments operate.

Formalization Note Entries are natural numbers with the constraint $a_{ij}\in\{0,1\}$ imposed explicitly, and the matrix square is written entrywise as $\sum_k a_{ik}a_{kj}$; the right-hand side uses if i = j then 12 else 0 for the term $12\,[i=j]$.