The Laplacian spectral excess theorem for distance-regular graphs

The spectral excess theorem states that, in a regular graph G, 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 G), and G 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 G

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

An application of Hoffman graphs for spectral characterizations of graphs

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the $2$-clique extension of the $(t+1)\times(t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough

Euclidean Distances, soft and spectral Clustering on Weighted Graphs

We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow data, properly conditioned, illustrate the procedure as well as visualization aspects.Comment: accepted for presentation (and further publication) at the ECML PKDD 2010 conferenc

The spectral excess theorem for graphs with few eigenvalues whose distance- 2 or distance-1-or-2 graph is strongly regular

We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.Research of C. Dalfó and M. A. Fiol is partially supported by Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) under project 2017SGR1087. Research of J. Koolen is partially supported by the National Natural Science Foundation of China under project No. 11471009, and the Chinese Academy of Sciences under its ‘100 talent’ programme. The research of C. Dalfó has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922