Homotopy in \(Q\)-polynomial distance-regular graphs
A connected graph \(\Gamma= (X,R)\) with diameter \(d\geq 1\) is called distance-regular if for all integers \(h\), \(i\), \(j\) such that \(0\leq h\), \(i,j\leq d\), and for all \(x,y\in X\) with \(\partial(x,y)= h\) the number \(p^h_{ij}= |\{z\in X\mid\partial(x, z)= i\text{ and }\partial(y,z)= j\}|\) depends only on \(h\), \(i\), \(j\) and not on \(x\) or \(y\). The integers \(p^h_{ij}\) are called the intersection numbers of \(\Gamma\). Since in a distance-regular graph \(\Gamma\) we have \(p^0_{11}= k\), \(\Gamma\) must be regular with valency \(k\). Considering the concept of the Bose-Mesner algebra of a distance-regular graph \(\Gamma\) we say that \(\Gamma\) is \(Q\)-polynomial if \(\Gamma\) has at least one \(Q\)-idempotent. The author proves that if \(\Gamma= (X,R)\) is a \(Q\)-polynomial distance-regular graph with diameter \(d\geq 3\) and valency \(k\geq 3\), then the intersection number \(p^3_{12}\) is at least 2 and the girth of \(\Gamma\) is at most 6. The author also defines the concept of a pseudoquotient graph and proves that if \(\Gamma\) is not a pseudoquotient graph, then any cycle in \(\Gamma\) can be decomposed into cycles of length at most 6.
scientific article; zbMATH DE number 1538206