On the primitive idempotents of distance-regular graphs

Let Γ denote a distance-regular graph with diameter d3. Let E, F denote nontrivial primitive idempotents of Γ such that F corresponds to the second largest or the least eigenvalue. We investigate the situation that there exists a primitive idempotent H of Γ such that EF is a linear combination of F and H. Our main purpose is to obtain the inequalities involving the cosines of E, and to show that equality is closely related to Γ being Q-polynomial with respect to E. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.     

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ₁ (resp., θ_D) we show that (θ₁+1)(θ_D+1)?-b₁ holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.     

E_1 ~oE_d型距离正则图的关于余弦序列的不等式

给出了E1 Ed型距离正则图的关于余弦序列的不等式.     

Antipodal distance-regular graphs of diameter four and five

An antipodal distance-regular graph of diameter four or five is a covering graph of a connected strongly regular graph. We give existence conditions for these graphs and show for some types of strongly regular graphs that no nontrivial covers exist. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 69–77, 1999     

Distance-regular graphs with complete multipartite <Emphasis Type="Italic">μ</Emphasis>-graphs and AT4 family

Let Γ be an antipodal distance-regular graph of diameter 4, with eigenvalues . Then its Krein parameter vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regular with nontrivial eigenvalues and . When this is the case, the intersection parameters of Γ can be parametrized by p, q and the size of the antipodal classes r of Γ. Let Γ be an antipodal tight graph of diameter 4, denoted by AT4 (p, q, r), and let the μ-graph be a graph that is induced by the common neighbours of two vertices at distance 2. Then we show that all the μ-graphs of Γ are complete multipartite if and only if Γ is AT4(sq,q,q) for some natural number s. As a consequence, we derive new existence conditions for graphs of the AT4 family whose μ-graphs are not complete multipartite. Another interesting application of our results is also that we were able to show that the μ-graphs of a distance-regular graph with the same intersection array as the Patterson graph are the complete bipartite graph K _4,4. Authors were supported in part by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant # R11-1999-054) and in part by the Slovenian Ministry of Science, while the first author was visiting the Combinatorial and Computational Mathematics Center at POSTECH, and while the second author was visiting the IMFM at the University of Ljubljana.