arXiv:2611.00126v1 Announce Type: new Abstract: Koolen, Hayat and Iqbal asked for the connected graphs whose distance matrix has exactly three distinct eigenvalues, the middle one being $0$. The examples they gave, namely the Johnson graphs, the Hamming graphs and the halved cubes, are distance-regular. In this note, we show that the distance spectrum of such a graph is $\{\rho,0,-c\}$ with integers $c\ge2$ and $c\mid\rho$. For $c=2$ the graphs are exactly the cocktail party graphs, and for $c=3$ they are exactly the Cartesian product $K_3\,\square\,K_3$ of two triangles, the Petersen graph and its complement. We also show that such a graph need not be transmission-regular: Seidel switching the lattice graph $K_6\,\square\,K_6$ with respect to a line and the triangular graph $T(11)$ with respect to a maximal clique of size $10$ gives graphs on $36$ and $55$ vertices with distance spectra $\{54,0^{(26)},(-6)^{(9)}\}$ and $\{81,0^{(45)},(-9)^{(9)}\}$. Among the graphs of diameter $2$ obtained by switching a strongly regular graph with smallest eigenvalue $-2$ with respect to one part of an equitable partition, these two are the only ones that have exactly three distance eigenvalues and are not transmission-regular. We also observe that the transmission-regular graphs with exactly three distinct distance eigenvalues, the smallest one being $-3$, are exactly the complements of the strongly regular graphs with smallest eigenvalue $-2$ other than the cocktail party graphs.
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