Quasi-Spectral Characterization of Strongly Distance-Regular Graphs

A graph with diameter d is strongly distance-regular if is distanceregular and its distance-d graph d is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with d distinct eigenvalues), in terms of the eigenvalues, the sum of the multiplicities corresponding to the eigenvalues with (non-zero) even subindex, and the harmonic mean of the degrees of the distance-d graph. AMS subject classifications. 05C50 05E30 1 Preliminaries Strongly distance-regular graphs were recently introduced by the author [9] by combining the standard concepts of distance-regularity and strong regularity. A strongly distance-regular graph is a distance-regular graph (of diameter d, say) with the property that the distance-d graph d--where two vertices are adjacent whenever they are...

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Miguel Angel Fiol
• Keywords: Antipodal Distanceregular Graphs, COMBINATORICS 2000, Distance-d Graph, Distance-regular Graphs