International Journal of Combinatorics
Volume 2011 (2011), Article ID 101928, 15 pages
Research Article

Binary Representations of Regular Graphs

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 17 January 2011; Accepted 2 August 2011

Academic Editor: Hajo Broersma

Copyright © 2011 Yury J. Ionin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For any 2-distance set 𝑋 in the n-dimensional binary Hamming space 𝐻 𝑛 , let Γ 𝑋 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 Γ 𝑋 , where 𝑋 is a 2-distance set lying on a sphere in 𝐻 𝑛 . 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.