Binary Representations of Regular Graphs

For any 2-distance set X in the n-dimensional binary Hamming spaceHn, let ΓX be the graph with X as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in X. The binary spherical representation number of a graph Γ, or bsr Γ , is the least n such that Γ is isomorphic to ΓX , where X is a 2-distance set lying on a sphere in Hn. It is shown that if Γ is a connected regular graph, then bsr Γ ≥ b − m, where b is the order of Γ and m is the multiplicity of the least eigenvalue of Γ, and the case of equality is characterized. In particular, if Γ is a connected strongly regular graph, then bsr Γ b − m if and only if Γ is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.