Some spectral characterizations of strongly distance-regular graphs (Q2726709)

An \(n\)-vertex graph \(\Gamma\) with diameter \(d\) is strongly distance-regular with parameters \((n,k;a,c)\) if \(\Gamma\) is strongly distance-regular and its dictance-\(d\) graph \(\Gamma_d\) is strongly regular with parameters \(k=|\Gamma_d|\), \(\lambda=a\) and \(\mu=c\). Given a mesh of real points \({\mathcal M}=\{x_0>x_1>\dots >x_m\}\) we set \(\pi_i({\mathcal M})=\prod_{j=0,j\neq i}^m |x_i-x_j|\) and \(\Pi_k({\mathcal M})=\sum_{i\text{ even}}(x_i^k/\pi_i)\). If \({\mathcal M}\) is a set of different eigenvalues of a graph \(\Gamma\) then we write \(\pi_i\) and \(\Pi_i\) as abbreviations of \(\pi_i({\mathcal M})\) and \(\Pi_i({\mathcal M})\). Let \(\Gamma\) be an \(n\)-vertex graph with spectrum \(\text{Sp}(\Gamma)= \{\theta_0^{m_0},\theta_1^{m_1},\dots ,\theta_d^{m_d}\}\). Fiol and Garriga proved that \(\Gamma\) is distance-regular if and only if its dictance-\(d\) graph \(\Gamma_d\) is regular with degree NEWLINE\[NEWLINEk=n(\sum_{i=0}^d \frac{\pi_0^2}{m_i\pi_i^2})^{-1}.NEWLINE\]NEWLINE The main result of this paper is Theorem 3.3: A distance-regular graph \(\Gamma\) with \(n\) vertices and spectrum \(\text{Sp}(\Gamma)=\{\theta_0^{m_0},\theta_1^{m_1},\dots ,\theta_d^{m_d}\}\) is \((n,k;a,c)\)-strongly distance-regular if and only if NEWLINE\[NEWLINEm_i=\frac{\pi_0}{\pi_i} \frac{\rho (n-1)+\sqrt{(n-1)(\rho^2(n-1)+4k (n+\rho-k))}}{2(n+\rho-k)} \qquad (i \text{ odd}),NEWLINE\]NEWLINE NEWLINE\[NEWLINEm_i=\frac{\pi_0}{\pi_i}\frac{\rho (1-n)+\sqrt{(n-1)(\rho^2(n-1)+4k (n+\rho-k))}}{2(n+\rho-k)} \qquad (i\text{ even and }i\neq 0),NEWLINE\]NEWLINE where \(\rho=a-c\). A consequence of Theorem 3.3 is the following necessary condition for strongly distance regularity. Let \(\Gamma\) be a strongly distance-regular graph with different eigenvalues \({\mathcal M}=\{\theta_0>\theta_1>\dots >\theta_d\}\). Then NEWLINE\[NEWLINE\Pi_1({\mathcal M}+1)=\sum_{i \text{ even}}\frac{\theta_i+1}{\pi_i({\mathcal M})} \sum_{i \text{ odd}}\frac{\theta_i+1}{\pi_i({\mathcal M})}.NEWLINE\]NEWLINE For \(d=3\) the above condition yields \(\theta_2=-1\).