Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: An Eigenvalue Characterization of Antipodal Distance-Regular Graphs. | An Eigenvalue Characterization of Antipodal Distance-Regular Graphs M. A. Fiol Departament de Matemàtica Aplicada i Telematica Universitat Politàcnica de Catalunya Jordi Girona 1-3 Màdul C3 Campus Nord 08034 Barcelona Spain email fiol@ Submitted July 19 1997 Accepted November 14 1997. Abstract Let r be a regular connected graph with n vertices and d 1 distinct eigenvalues. As a main result it is shown that r is an r-antipodal distanceregular graph if and only if the distance graph rd is constituted by disjoint copies of the complete graph Kr with r satisfying an expression in terms of n and the distinct eigenvalues. AMS subject classifications. 05C50 05E30 1 Introduction The core of spectral graph theory is to describe the properties of a graph by its spectrum and find conditions that cospectral graphs may not share. For instance consider the following question Can we see from the spectrum of a graph with diameter D say whether it is distance-regular Since a long time it was known that the answer to this question is yes when D 2 and not if D 4. Then on the basis of these results it had been conjectured cf. Cvetkovic Doob and H. Sachs 5 that the answer is also yes for D 3 but recently Haemers 19 disproved the conjecture constructing some counterexamples. So in general the spectrum is not sufficient to assure distance-regularity and if we want to go further we must require the graph to satisfy some additional conditions. In this direction Van Dam and Haemers 8 showed that in the case D 3 such a condition could be the number nd of vertices THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 1997 R30 2 at extremal distance D d where d 1 is the number of distinct eigenvalues from each vertex. Independently Garriga Yebra and the author 13 settled the case nd 1 for any value of D that is the case of 2-antipodal distance-regular graphs. Finally Garriga and the author 11 solved the general case characterizing distance-regular graphs as those regular graphs whose number of
