Spin Models and Almost Bipartite 2-Homogeneous Graphs

Abstract

A connected graph of diameter $d$ is said to be almost bipartite if it contains no cycle of length $2\ell + 1$ for all $\ell < d$. An almost bipartite distance-regular graph $\Gamma = (X, E)$ is 2-homogeneous if and only if there are constants $\gamma_1, \ldots, \gamma_d$ such that $|\Gamma_{i-1}(u) \cap \Gamma_1(x) \cap \Gamma_1(y)| = \gamma_i$ holds for all $u \in X$ and for all $x, y \in \Gamma_i(u)$ with $\partial(x, y) = 2$ $\ (i=1, \ldots, d)$.

In this paper, almost bipartite 2-homogeneous distance-regular graphs are classified. This determines triangle-free connected graphs affording spin models (for link invariants) with certain weights.