This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc.

The *determinant* of a simple graph is the determinant of its adjacency matrix which is, of course, a symmetric binary matrix with zero-diagonal.

* Question:* Which graph has the largest value of $|\det(A)|$ over all simple undirected graphs on $n$ vertices?

(We could ask separately for the graphs with * minimum* determinant, which will be some negative number, and those of

*determinant, but by taking the absolute value, this just becomes a question of which graphs have the*

**maximum***determinant.)*

**most extreme**For small numbers of vertices, we can just find the answer almost by hand:

- On $2$ vertices, the winner is the complete graph $K_2$ with spectrum $\{-1,1\}$, so determinant $-1$, of absolute value $1$.
- On $3$ vertices, the winner is $K_3$ which has absolute value of determinant equal to $2$
- On $4$ vertices, the winner is $K_4$ with value $3$.
- On $5$ vertices, there are two winners (both with value $4$), namely $K_5$ and the "bowtie" graph (two triangles sharing a common vertex).

Sequence so far is $1$, $2$, $3$, $4$ (don't bother looking this up in OEIS).

For larger numbers of vertices, we turn to the computer and discover that on $6$ vertices, the maximum value is $7$, and this is realised by two graphs, namely a triangle and a $4$-clique sharing a vertex, and two $4$-cliques sharing an edge.

From there, the sequence continues: 12, 28, 128, 256, 576 for 7, 8, 9, 10 and 11 vertices respectively, with between 2 and 7 graphs achieving the maximum for each of these values.

So now I do have enough to look up in the OEIS, but there are no matches.

The problem of finding the maximum determinant of a (0,1)-matrix, often called the Hadamard maximal determinant problem has been extensively studied, and there are lots of bounds, and constructions of extremal matrices etc. (There is a mapping between (0,1)-matrices and (-1,1)-matrices which changes the determinant predictably and when they exist, Hadamard matrices are the winners.)

So lots is known, and it is summarised in sequence A003432 on the OEIS which gives exact values up to $n=20$.

Just for comparison, for $n = 6$ to $n=11$, we have

- 7, 12, 28, 128, 256, 576 (for graphs)
- 9, 32, 56, 144, 320, 1458 (for (0,1)-matrices)

From several sources, including the OEIS, I have been advised to check a website attributed to Will Orrick at http://www.indiana.edu/~maxdet/ but this appears to be offline or removed or something, and nor could I find his email address in the directory for that university.

So my question remains: what is known about maximal determinant (in absolute value) of the adjacency matrix of a simple graph on $n$ vertices?

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