The spectra of complementary subgraphs in a strongly regular graph
Let \(C=xI-A\), where \(A\) is the adjacency matrix of a strongly regular graph \(G\) with parameters \((v,k,\lambda,\mu)\). Then \[ q(x)C^{-1}=A+(x+\mu-\lambda)I+\mu/(x-k)J, \] where \(q(x)=x^2+(\mu-\lambda)x+(\mu-k)\). Let \(r>0\), \(s\) be a nonprincipal eigenvalue of \(A\) with multiplicities \(f,g\). One may take the indexing set for rows and colomns of any \(n\times n\) matrix \(M\) to be \([n]=\{1,2,\dots,n\}\). If \(T\subseteq [n]\), let \(M[T]\) denote the submatrix of \(M\) whose row and colomn indices are \(T\). Also put \(\overline{T}=[n]-T\), and \(| M|\) denote the determinant of \(M\). Theorem 1. Let \(G\) be a strongly regular graph with adjacency matrix \(A\) and characteristic matrix \(C=xI-A\). For any set \(T\) of \(t\) vertices, \[ | C[\overline{T}]=q(x)^{-t}| C|\;| A[T]+(x+\mu-\lambda)I_t+\mu/(x-k)J_t|, \] where \(q(x)=x^2+(\mu-\lambda)x+(\mu-k)\). Corollary 3. For any vertex \(p\), \[ | C[G_2(p)]=(x-k+\mu)(x-r)^{f-k}(x-s)^{g-k}\prod_{\omega}(x+\mu-\lambda+ \omega), \] where \(\omega\) ranges over the eigenvalues of \(A[G_1(p)]\) with \(\omega=\lambda\) omitted once. An antipodal distance-regular graph of diameter \(3\) is called \((n,r,c_2)\)-cover, if \(n\) is the number of antipodal classes (called fibres), \(r\) is the size of each fibre and \(c_2\) is the number of common neighbours to any pair of vertices at distance \(2\). The eigenvalues of any \((n,r,c_2)\)-cover are \(n-1\), \(-1\), \(\theta\), \(\tau\) with multiplicities \(1\), \(n-1\), \(m(\theta)=n(r-1)\tau/(\tau-\theta)\), \(m(\tau)=n(r-1)\theta/(\theta-\tau)\). Theorem 7. Let \(G\) be a strongly regular graph with parameters \((v,k,\lambda,\mu)\) such that for some vertex \(p\), subgraph \(G_2(p)\) is \((n,r,c_2)\)-cover. Then \(\theta\), \(\tau\) are eigenvalues of \(G\) with multiplicities \[ f=(\tau(v-1)+k)/(\tau-\theta), \qquad g=(\theta(v-1)+k)/(\theta-\tau), \] and \[ \prod_{\omega}(x+\mu-\lambda+\omega)=(x+1)^{n-1} (x-\theta)^{m(\theta)-f+k}(x-\tau)^{m(\tau)-g+k}, \] where \(\omega\) ranges over the eigenvalues of \(A[G_1(p)]\) with \(\omega=\lambda\) omitted once. In particular, the inequalities \(m(\theta)-f+k\geq 0\) and \(m(\tau)-g+k\geq 0\) hold. For example, the following sets \((n,r,c_2)\) appear to be feasible (\(\gamma=c_2(c_2-1)/(n-c_2)\) is an integer, etc.) but are killed by Theorem 7: \((16,2,6)\), \((33,3,9)\), \((36,4,8)\), \((45,3,12)\), \((55,5,10)\), \((56,5,12)\).
scientific article; zbMATH DE number 1199937