I would like to characterize the self-dual strongly regular graphs in terms of their eigenvalues.

The entries of the dual character table of a symmetric coherent algebra can be expressed in terms of the entries its character table:

  • Suppose that \(W\) is a symmetric coherent algebra of order \(n\) and dimension \(d\).

    Let \(P \in \operatorname{Mat}_{d \times d}(\mathbb{C})\) denote the character table of \(W\) with dual character table \(Q \in \operatorname{Mat}_{d \times d}(\mathbb{C})\) relative to orderings \(\{ A_{j} : 1 \leq j \leq d \}\) and \(\{ E_{j} : 1 \leq j \leq d \}\) on its sets of Schur-primitive and primitive matrices respectively.

    Additionally, let \(v_{j}\) denote the valency of \(A_{j}\) and \(m_{j}\) denote the rank of \(E_{j}\) for all \(1 \leq j \leq d\).

    Define the trace inner product \(\langle \cdot, \cdot \rangle : \operatorname{Mat}_{n \times n}(\mathbb{C}) \to \operatorname{Mat}_{n \times n}(\mathbb{C})\) on \(\operatorname{Mat}_{n \times n}(\mathbb{C})\):

    \[\langle A, B \rangle := \operatorname{tr}\left ( B^{*} A \right ) = \operatorname{sum} \left (\overline{B} \circ A \right )\]

    .

    \[\implies v_{j} Q_{i, j} = \langle E_{i}, A_{j} \rangle = m_{i} P_{j, i} \text{ for all } 1 \leq i, j \leq d\]

    \(\implies Q = R \circ P^{T}\), defining \(R \in \operatorname{Mat}_{d \times d}(\mathbb{C})\), \(R_{i,j} := \frac{m_{i}}{v_{j}}\).

Now, suppose that \(G\) is a (connected) strongly regular graph of order \(n \in \mathbb{N}\) and valency \(k \in \mathbb{N}\).

\(\implies\) \(G\) has three distinct eigenvalues \(\{ k, \lambda, \mu \}\) for some \(\lambda, \mu \in \mathbb{R}\).

Let \(j\) denote the multiplicity of \(\lambda\) as an eigenvalue of \(G\).

Following my previous point, it is then possible to derive that

\[P := \begin{pmatrix} 1 & 1 & 1 \\ k & \lambda & \mu \\ {n - 1 - k} & {-1 -\lambda} & {-1 - \mu } \end{pmatrix}\]

is a character table for the adjacency algebra of \(G\)

with dual character table \(Q := \begin{pmatrix} 1 & 1 & 1 \\ j & \lambda^{'} & \mu^{'} \\ {n - 1 - j} & {-1 - \lambda^{'}} & {-1 - \mu^{'}} \end{pmatrix}\).

letting \(\lambda^{'} := \left ( \frac{j}{k} \right ) \lambda\) and \(\mu^{'} := \left ( \frac{j}{n - 1 - k} \right ) \left ( -1 - \lambda \right )\).

Recall that \(PQ = nI\).

\[\implies \begin{cases} k + \left ( \lambda^{'} \right ) \lambda + \left ( -1 - \lambda^{'} \right ) \mu = n \\ k + \left ( \mu^{'} \right ) \lambda + \left ( -1 - \mu^{'} \right ) \mu = 0 \end{cases}\] \[\implies \left ( \lambda - \mu \right ) \left ( \lambda^{'} - \mu^{'} \right ) = n\]

We also have from this that: \(k + (j) \lambda + (n - 1 - j) \mu = 0 = j + (k) \lambda^{'} + (n - 1 - k) \mu^{'}\).

I would then propose that \(G\) is self-dual if and only if \(\left ( \lambda - \mu \right )^{2} = n\):

  • Assume that \(\left ( \lambda - \mu \right )^{2} = n\)

    \[\implies \left ( \lambda - \mu \right )^{2} = \left ( \lambda - \mu \right ) \left ( \lambda^{'} - \mu^{'} \right )\] \[\left ( \lambda - \mu \right ) = \left ( \lambda^{'} - \mu^{'} \right )\]

    [ since \(\lambda\) and \(\mu\) are distinct ]

    \[k + (j) \lambda + (n - 1 - j) \mu = 0 = j + (k) \lambda + (n - 1 - k) \mu\]

    [ following my previous point ]

    \[\implies (j - k)\left ( 1 - \lambda + \mu \right ) = 0\] \[\implies j = k\]

    [ since \(n > 1\) and \(n = 1\) otherwise ]

    \[\implies Q = \begin{pmatrix} 1 & 1 & 1 \\ k & \lambda & \left ( \frac{k}{n - 1 - k} \right ) \left ( -1 - \lambda \right ) \\ {n - 1 - k} & \left ( \frac{n - 1 - k}{k} \right ) \mu & {-1 - \mu} \end{pmatrix}\] \[\implies \left ( e_{1}^{T} P \right ) \left ( Q e_{2} \right ) = 1 + \lambda + \left ( \frac{n - 1 - k}{k} \right ) \mu = 0\] \[\implies P = Q\]

    and hence \(G\) is self-dual.

It is conversely clear that \(\left ( \lambda - \mu \right )^{2} = n\) if \(G\) is self-dual (and hence \(P = Q\)).

So, \(G\) is indeed self-dual if and only if \(\left ( \lambda - \mu \right )^{2} = n\).

A first example of a family of self-dual strongly regular graphs is then the conference graphs; I will return to work towards completely classifying the self-dual strongly regular graphs in a future post.