The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.
|Journal||Linear Algebra and its Applications|
|Early online date||24 Jun 2014|
|Publication status||Published - 1 Oct 2014|
- Distance-regular graphs
- spectral excess theorem
- Laplacian spectrum
- orthogonal polynomials