Distance-regular graphs with an eigenvalue \(-k \theta \leq 2-k\)

Sejeong Bang¹

¹Yeungnam University

The electronic journal of combinatorics, Tome 21 (2014) no. 1

Résumé

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)$.

Détail

DOI : 10.37236/3304

Classification : 05C50, 05C12
Mots-clés : distance-regular graph, girth, smallest eigenvalue, folded \((2D + 1)\)-cube