Star partitions and regularity in graphs (Q1899409)

Let \(G\) be a graph with the vertex set \(X= \{1, 2,\dots, n\}\) and let \(e_1, e_2,\dots, e_n\) be a standard basis of \(\mathbb{R}^n\). Let \(\mu_1, \mu_2,\dots, \mu_m\) be distinct eigenvalues of (the adjacency matrix of) \(G\). For each \(i\in \{1, 2,\dots, m\}\) let \(P_i\) be the orthogonal projection of \(\mathbb{R}^n\) onto the eigenspace of \(\mu_i\). A partition \(X_1\dot\cup\cdots \dot\cup X_m\) of \(X\) is called a star partition of \(G\) if the set \(\{P_i e_j\mid j\in X_i,\;i= 1,2,\dots, m\}\) is a basis of \(\mathbb{R}^n\). The author investigates regular graphs with a star partition \(X_1\dot\cup X_2\dot\cup\cdots \dot\cup X_m\) such that \(G- X_i\) is regular for some \(i\). A partial classification of cubic graphs with this property is given.