Abstract
It is known that bipartite distance-regular graphs with diameter $D\geq 3$, valency $k\geq 3$, intersection number $c_2\geq 2$ and eigenvalues $k = \theta_0 > \theta_1 > \cdots > \theta_D$ satisfy $\theta_1\leq k-2$ and thus $\theta_{D-1}\geq 2-k$. In this paper we classify non-complete distance-regular graphs with valency $k\geq 2$, intersection number $c_2\geq 2$ and an eigenvalue $\theta$ satisfying $-k< \theta \leq 2-k$. Moreover, we give a lower bound for valency $k$ which implies $\theta_D \geq 2-k$ for distance-regular graphs with girth $g\geq 5$ satisfying $g=5$ or $ g \equiv 3~(\operatorname{mod}~4)$.
Highlights
Let Γ be a distance-regular graph with diameter D 3 and eigenvalues k = θ0 > θ1 > · · · > θD
It is known that bipartite distance-regular graphs with diameter D 3, valency k 3, intersection number c2 2 and eigenvalues k = θ0 > θ1 > · · · > θD satisfy θ1 k − 2 and θD−1 2 − k
We give a lower bound for valency k which implies θD 2 − k for distance-regular graphs with girth g 5 satisfying g = 5 or g ≡ 3
Summary
Any non-complete bipartite distanceregular graph Γ with valency k 2, intersection number c2 2 and an eigenvalue θ with −k < θ 2 − k satisfies θ = 2 − k and Γ is either the cycle of length four or the Hamming D-cube by 2 − k −θ1 θ 2 − k and [2, Theorem 4.4.11]. In the following theorem we classify non-complete distance-regular graphs with valency k 2, intersection number c2 2 and an eigenvalue θ satisfying −k < θ 2 − k. The folded n-cube (n = 6) is uniquely characterized by its intersection array (cf [2, Theorem 9.2.7]) It follows by Theorem 1 that a distance-regular graph with D 3, k 3, c2 2 and an eigenvalue θ satisfying −k < θ 2 − k is either the Hamming D-cube or the folded (2D + 1)-cube.
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