Let F be a distance-regular graph with valency k (k >I 2) and diameter at least 2, and denote by ;t 1 and 2%~m the second largest and least eigenvalue of F, respectively. Assume the multiplicity m( )O of some eigenvalue ;~ ( )~ :/: k) of F satisfies m( Z ) < k. Then ;~ = Z 1 or )'rot. and either (i) 3. is an integer such that 1 + ), divides the intersection number b x, or (ii) )~ and bx(1 + 3.) -1 are algebraic integers, m(Zl) = m(3.mi.), and )L 1 and ~tmi ~ are algebraic conjugates over Q.
We also obtain bounds on the second largest and least eigenvalue of subgraphs of F induced by vertex neighborhoods.
Let A be a symmetric n x n matrix with real coefficients acting on the vector space V = R n, and let Sp(A) be the set of distinct eigenvalues of A. For 3. e Sp(A) let Vx(A) be the 3.-eigenspace of A, with dimension denoted by m(3.). Let 3.0, 3.1, and 3.mi, be the largest, second largest, and least element of Sp(A). Now let F be a finite, undirected, loopless graph, regular with some valency k, with vertex set VF = {vl, v2, β’ β’ β’, v.}. The n x n adjacency matrix A = A(F) of F satisfies AΒ°=fl[o ifvi and vj are adjacent, otherwise, and we write Sp(F)= Sp(A). It is well known (see for example Biggs [2, p. 14]) that the regularity of F implies 3.o = k, with re(k) given by the number of connected components of F. We refer the reader to [2] for definitions and basic facts about graphs. Now assume F is connected with diameter d. Denoting by Fj(u) (0 ~<j <~ d) the set of vertices in VF at distance j from any u e VF, we say F is distance-regular with intersection matrix lao bo O Cl al bl C2 0 bd-1