We consider a primitive distance-regular graph
Γ
with diameter at least 3. We use the intersection numbers of
Γ
to find a positive semidefinite matrix
G
with integer entries. We show that
G
has ...determinant zero if and only if
Γ
is
Q
-polynomial.
We prove that a distance-regular graph with intersection array {22,16,5;1,2,20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local ...graph. Then we construct a partial linear space from the local graph to display the contradiction.
In this paper we consider a distance-regular graph \(\Gamma\). Fix a vertex \(x\) of \(\Gamma\) and consider the corresponding subconstituent algebra \(T\). The algebra \(T\) is the ...\(\mathbb{C}\)-algebra generated by the Bose-Mesner algebra \(M\) of \(\Gamma\) and the dual Bose-Mesner algebra \(M^*\) of \(\Gamma\) with respect to \(x\). We consider the subspaces \(M, M^*, MM^*, M^*M, MM^*M, M^*MM^*, \dots\) along with their intersections and sums. In our notation, \(MM^*\) means \(Span\{RS|R\in M, S\in M^*\}\), and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to \(MM^*+M^*M\). For each subspace \(U\) shown in this part of the diagram, we display an orthogonal basis for \(U\) along with the dimension of \(U\). For an edge \(U\subseteq W\) from this part of the diagram, we display an orthogonal basis for the orthogonal complement of \(U\) in \(W\) along with the dimension of this orthogonal complement.
We consider a primitive distance-regular graph $\Gamma$ with diameter at
least $3$. We use the intersection numbers of $\Gamma$ to find a positive
semidefinite matrix $G$ with integer entries. We ...show that $G$ has determinant
zero if and only if $\Gamma$ is $Q$-polynomial.
We consider a primitive distance-regular graph \(\Gamma\) with diameter at least \(3\). We use the intersection numbers of \(\Gamma\) to find a positive semidefinite matrix \(G\) with integer ...entries. We show that \(G\) has determinant zero if and only if \(\Gamma\) is \(Q\)-polynomial.
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